Minimal transitive-subset conjecture for permutation encoding of input-symmetric degraded broadcast channels

Let $k$ be the alphabet size of the channel input, and let $l_s$ be the alphabet size of the code for User 2 in a permutation encoding approach. For an input-symmetric degraded broadcast channel, let $\mathcal{G}_{T_{YX},T_{ZX}}$ denote the relevant family of matrices, and let $\{G_1,\ldots,G_{l_s}\}$ be a smallest transitive subset of this family. Minimal transitive-subset conjecture. The code alphabet size satisfies $l_s=k$, and

$$\sum_{j=1}^{l_s}G_j=\boldsymbol{1}\boldsymbol{1}^T.$$

This conjecture concerns whether permutation encoding achieves the stated alphabet-size bound for User 2 on every input-symmetric degraded broadcast channel. The supplied text does not state whether the claim has been proved or disproved.