The J12 Conjecture. (Can a specific causal structure imply specific distance ratios ?)

Since recently there appear widened oportunities to search for counterexamples to specifically stated mathematical conjectures in general, I'd like to derive and present a suitable conjecture in the first place, arising from the context of this blog. For now, this leads me to "the J₁₂ conjecture" to be presented in the following. The name invokes the Johnson solid J₁₂ which is listed, by convention, as the twelfth (among a total of 92 polyhedra), and which is also (perhaps more descriptively, but certainly less conveniently) known as "regular triangular bipyramid"; encompassing 5 vertices, where 9 pairs of vertices (i.e. all pairs "but one") are connected by edges of equal length. (The remaining vertex pair, which is not visibly connected by an edge, has a distance from each other $\sqrt{8/3}$ of the length of all edges.)

1. Concise, high-level formulation of the J₁₂ conjecture:

In (3+1)-dimensional flat spacetime, if five timelike worldlines are permanently at rest wrt. each other, and specificly with distance ratios among them just as among the five vertices of a (non-rotating, regular) J₁₂ polyhedron,

​

then these five participants (in the following sketches shown as green dots) find a very specific permanent causal structure among each other; namely:
- each of the three vertices of the J₁₂ polyhedron center plane (i.e. the horizontal plane in the sketches) finds ping coincidences wrt. all four other vertices;

- and both remaining vertices (i.e. the two tips of the J₁₂ polyhedron; in the sketches one at the top, the other one at the bottom) each find ping coincidences wrt. all three vertices of the J₁₂ center plane.

Moreover, the light-like paths which connect any two of those J₁₂ vertices (which appear in the sketches as projected on any one of the edges between vertices) make up distinct photon-2-surfaces (specificly: photon-2-ribbons, i.e. with with the time-like world lines of the vertices as borders), consisting explicitly of events in which certain pairs of those light-like paths intersect. Certain sets of those intersection events are in turn identifiable as the time-like world lines of participants who each constitute "the midpoint between" two vertices of the J₁₂ polyhedron. (The following sketches show nine of these "midpoints between" vertices as blue dots; the nine "external midpoints". The "midpoint between" the two tip vertices of the J₁₂ polyhedron can be identified as well, but is not of immediate concern for the J₁₂ Conjecture; and, therefore, is not shown in the sketches.)

The nine "external midpoints" also find very specific permanent causal structure among each other, as well as wrt. the five vertices of the J₁₂ polyhedron. Attempting to completely list the corresponding ping-coincidence relations (in addition to those listed above, found by the five vertices among each other):
- each of the nine external midpoints finds ping coincidence wrt. its 2 respective adjacent vertices;
- each of the three midpoints of edges in the J₁₂ center plane also finds this exact same ping coincidence wrt. six other external midpoints; exemplary:

- each of the six midpoints of edges between J₁₂ center plane and J₁₂ tips also finds ping coincidence wrt. four other external midpoints exactly same as wrt. its two adjacent J₁₂ vertices; exemplary:

- each of the three vertices in the J₁₂ center plane finds ping coincidence wrt. its four adjacent external midpoints; with any two successive of these pings exactly the same as one ping wrt. any other vertex; exemplary:

- each of the two J₁₂ tip vertices finds ping coincidence wrt. its three adjacent external midpoints; with any two successive of these pings exactly the same as one ping wrt. any vertex in the J₁₂ center plane; exemplary:

All ping-coincidence relations listded so far and considered together may thus be called "in synchrony". There remain several "asynchronous" ping-coincidence relations which don't match with the above "synchronous" ones:
- of each of the three vertices in the J₁₂ center plane wrt. five external midpoints; exemplary:

- of each of the two J₁₂ tip vertices wrt. the three midpoints of edges in the J₁₂ center plane: exemplary

and also (separately, asynchronly) wrt. the three midpoints "adjacent to the other tip": exemplary

- of each of the three midpoints of edges in the J₁₂ center plane wrt. three non-adjacent vertices; exemplary:

- of each of the three midpoints of edges in the J₁₂ center plane wrt. the two midpoints of its "opposite edges": exemplary

and (finally)
- of each of the six midpoints of edges between J₁₂ center plane and J₁₂ tips wrt. the two midpoints of edges "to the other tip": exemplary

Now, the J₁₂ Conjecture claims that, in this particular case, the inference from given distance ratios (among five vertices constituting a regular J₁₂ polyhedron) to conclude causal structure (among these five J₁₂ vertices and their nine "external midpoints"; as listed above) can be inverted: five (distinct, non-intersecting) timelike worldlines in (3+1)-dimensional flat spacetime find the described permanent ping coincidences among each other, and go on to identify nine additional (distinct, non-intersecting) time-like worldlines such that all fourteen find the fully set of permanent ping-coincidence relations among each other not merely if, but only if they're all permanently at rest wrt. each other, with distance ratios as exhibited by the five vertices of a non-rotating, regular J₁₂ polyhedron. In other words, it is conjectured that the permanent causal structure exhibited by a non-rotating regular J₁₂ polyhedron, with its five vertices and nine external midpoints, as summarized by the above list of ping-coincidence relations, is sufficient to derive or conclude the distance relations permanently and exclusively.

2. Formulation of the J₁₂ Conjecture as elementary mathematics problem (determining functions in $\mathbb R^3$)