Minimum minimizing index conjecture for doubled regular odd prime-gons

Let $X_p$ be the doubled regular $p$-gon, where $p$ is an odd prime. The minimizing index $\operatorname{minind}(X_p)$ is the smallest positive integer $k$ such that $X_p$ admits a $1/k$-geodesic. Minimum minimizing index conjecture.

$$\operatorname{minind}(X_p)=2p.$$

The paper establishes that doubled regular odd $n$-gons admit a $1/2n$-geodesic and therefore gives the upper bound $\operatorname{minind}(X_n)\leq 2n$; it also rules out half-geodesics. The conjecture asserts that this upper bound is sharp when $n=p$ is an odd prime.