Khesin–Soloviev higher-dimensional pentagram Devron conjecture in dimension three

Let $A=(A_i)$ be a periodic sequence of $2n$ points in $\mathbb{RP}^3$, with $n\geq 4$. For each $i$, let the face at $i$ be the plane through three consecutive vertices, and let $F$ send $A$ to the sequence whose $i$th vertex is the intersection of the line through $A_{i-1}$ and $A_{i+1}$ with the plane through $A_{i-2},A_i,A_{i+2}$. Suppose there are points $P,Q$ such that the faces of $A$ pass through $P$ or $Q$ in alternating fashion.

Khesin–Soloviev higher-dimensional pentagram Devron conjecture. If $B=F^{n-3}(A)$, then $B_1,B_3,\ldots,B_{2n-1}$ are coplanar, and $B_2,B_4,\ldots,B_{2n}$ are also coplanar.

This is a conjectural Devron-property statement for the higher-dimensional pentagram map on general polygons in $\mathbb{RP}^3$, rather than only corrugated polygons. Its status is presented as experimental and unresolved in the source.