A fully supported local information maximum on the full two-shift

Let $(X,T)$ be the full two-shift, let $\alpha$ be the partition used to define information, and consider the space of one-step Markov measures on $X$. A measure is fully supported when every allowed cylinder has positive measure, and a local maximum is understood relative to this space. Fully supported local-maximum conjecture. There is a fully supported one-step Markov measure that is a local maximum of $\operatorname{Int}_\mu(X,T,\alpha)$ among all one-step Markov measures. The assertion is suggested by numerical computations in the source; no proof or resolution is supplied.