The extended local-permutation conjecture for block-shift matrices

Let $E(\mathfrak{t}_{\mathrm{loc}})$ be the relevant set of matrices, and let $\Pi_{\mathrm{loc}}$ be the subgroup of signed local permutations consisting of matrices

$$P_1\otimes\dots\otimes P_n,$$

where, for $k=1,\dots,n$,

$$P_k\in\left\{\pm I_2,\ \pm\begin{bmatrix}0&1\\-1&0\end{bmatrix}\right\}.$$

Let $P_{\rm out}$ be the displayed permutation matrix in the source, and let $\Pi_{\mathrm{out}}$ be the subgroup generated by inserting $P_{\rm out}$ in every possible tensor-factor position. Define

$$\Pi_{\mathrm{loc}}^{\mathrm{ex}}:=\Pi_{\mathrm{loc}}\cdot\Pi_{\mathrm{out}}.$$

A block-shift matrix is a matrix of the block-shift form described in the preceding classification.

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

This conjecture seeks the smallest natural enlargement of local signed permutations that accounts for all observed matrices with circular local $C$-numerical range; the proposed enlargement is generated by local permutations together with the displayed outer permutation in every tensor position.