Neighborliness conjecture for permutation polytopes of cyclic groups

Let $G=\langle g\rangle$ be the cyclic permutation group of order $d$, acting with $t$ orbits, and for $I\subseteq [t]$ let $d_I$ denote the relevant order associated with the union of the orbits indexed by $I$. Let $P(G)$ be the associated permutation polytope. Neighborliness conjecture. For $l\geq 1$, if $d_I=d$ for every $I\subseteq [t]$ with $|I|\geq \left\lceil\frac{t}{l+1}\right\rceil$, then $P(G)$ is $(l+1)$-neighborly: every subset of at most $l+1$ vertices of $P(G)$ is the vertex set of a face. The claim generalizes the preceding characterization of the complete vertex-edge graph, which is the case $l=1$; it was experimentally checked in many cases, while no proof or disproof is given here.