The Markov-chain extension of the most-informative Boolean function conjecture

Let $X^n\sim\operatorname{Unif}(\mathbb{H}^n)$, let $Y^n$ be obtained by passing $X^n$ through a binary symmetric channel with crossover probability $p\in[0,1]$, and let $V\in\mathbb{H}$ be binary with $X^n\to Y^n\to V$ forming a Markov chain. Define binary convolution by

$$a\ast b=a(1-b)+(1-a)b,$$

and let $H_2^{-1}:[0,1]\mapsto[0,1/2]$ be the inverse of binary entropy. The Markov-chain extension conjecture. One has

$$H_2\bigl(p\ast H_2^{-1}(1-I(V;Y^n))\bigr)\leq 1-I(V;X^n).$$

This extension preserves the degraded-channel structure and would provide a stronger route to the original conjecture. The paper gives no proof and treats it as open.