The conjecture that Theorem MainTechnical covers all rank-deficient permutations

Let $V$ be a generic Toeplitz matrix and let $P$ be a permutation matrix. Theorem MainTechnical applies to permutations satisfying one of the following conditions: there exists an integer $0\leq t\leq\frac{d}{2}$ such that

$$r_t:=\operatorname{rank}\big((P-J^t)_{t+1:n,1:n}\big)\leq d-2t,$$

or there exists an integer $-\frac{d}{2}\leq t<0$ such that

$$r_t:=\operatorname{rank}\big((P-(J^\top)^{-t})_{1:n+t,1:n}\big)\leq d+2t.$$

Rank-deficient permutation coverage conjecture. For $V$ generic Toeplitz, every permutation $P$ for which

$$\operatorname{rank}\left[V,\,PV\right]<2d$$

is covered by Theorem MainTechnical, that is, satisfies one of these conditions. The conjecture concerns the unresolved characterization of the permutations relevant to Question rank-P; it was verified by exhaustive computation for $n=2d$ and $d\leq5$, while a complete answer remains elusive.