Spin alignment conjecture for entropy minimization

Let $n$ be a positive integer, let $\sigma$ be a density operator, and let $M^c$ denote the complement of $M$ in $\{1,2,\cdots,n\}$. For a subset $M$, write $\sigma^{\otimes M^c}$ for the tensor product of copies of $\sigma$ on the tensor factors indexed by $M^c$, and similarly interpret $\lvert e_{k_0}\rangle\langle e_{k_0}\rvert^{\otimes M}$. Let $\{x_M\}_{M\subseteq \{1,2,\cdots,n\}}$ be a fixed probability distribution, and let $\lvert e_{k_0}\rangle$ be an eigenvector corresponding to the maximal eigenvalue of $\sigma$. Spin alignment conjecture. The entropy minimization problem in the stated spin-alignment setting is achieved at the state

$$\kappa=\sum_{M\subseteq \{1,2,\cdots,n\}}x_M\lvert e_{k_0}\rangle\langle e_{k_0}\rvert^{\otimes M}\otimes\sigma^{\otimes M^c}.$$

This conjecture identifies the entropy-minimizing state by aligning every tensor factor indexed by $M$ with a maximal-eigenvalue eigenvector of $\sigma$, while retaining $\sigma$ on the complementary factors. Its resolution is not established by the supplied material.