Prime characterization conjecture for the reduced Dressian of the cyclic matroid

Let $n\geqslant 7$ be an integer, and let ${\mathcal T}_n$ denote the cyclic matroid used in the paper. Write $\operatorname{\underline{Dr}}_{{\mathcal T}_n}$ for its reduced Dressian. Reduced Dressian conjecture.

1. The reduced Dressian $\operatorname{\underline{Dr}}_{{\mathcal T}_n}$ is one-dimensional if and only if $n$ is prime.
2. If $n=p$ is prime, then the one-dimensional fan $\operatorname{\underline{Dr}}_{{\mathcal T}_p}$ consists of exactly $\frac{p-1}{2}$ rays.

The conjecture concerns the combinatorial structure of the reduced Dressians of the cyclic matroids ${\mathcal T}_n$; the paper presents it after computations for several examples, and no resolution is supplied here.