The intrinsic-capacity decomposition conjecture

Let $W$ be a channel and let $\operatorname{dec}(W)$ denote its convex decompositions. For $\lambda\in\operatorname{dec}(W)$, write $\operatorname{C}_{11}(\lambda)$ for the corresponding capacity quantity, $\underline{\operatorname{IC}}_{11}(W)$ for the lower intrinsic capacity, and $\operatorname{\Gamma}_\lambda(1)$ and $\operatorname{\overline{\Gamma}}_W(1)$ for the associated rank probabilities. Intrinsic-capacity decomposition conjecture. If

$$\operatorname{C}_{11}(\lambda)=\underline{\operatorname{IC}}_{11}(W),$$

then

$$\operatorname{\Gamma}_\lambda(1)=\operatorname{\overline{\Gamma}}_W(1).$$

The conjecture asserts that a decomposition attaining the lower intrinsic capacity also attains the extremal rank probability at $1$; this would identify the decomposition needed to achieve the lower bound discussed in the surrounding results. Its resolution is not specified in the source.