Single-round KL contraction conjecture for the binary hypercube Gaussian-location channel

Let $Y$ be the output of the binary hypercube Gaussian-location channel, let $V$ denote its binary-hypercube input, and let $P_Y$ and $P_{V\mid Y}$ be the corresponding distributions and channel. Write $\eta_{\mathrm{KL}}(P_Y,P_{V\mid Y})$ for the distribution-dependent Kullback–Leibler contraction coefficient, and let $\mathrm{SNR}$ denote the channel's signal-to-noise ratio. Assume $\mathrm{SNR}\le 1$. Single-round KL contraction conjecture. There is an absolute constant $C$ such that

$$\eta_{\mathrm{KL}}(P_Y,P_{V\mid Y})\le C\,\mathrm{SNR}.$$

Under this conjecture, for a message $M$ and side information $U$ satisfying the setup in the paper with message budget $B$, one obtains $I(V;M\mid U)\le C\,\mathrm{SNR}\cdot B$; equivalently, the single-round case $T=1$ of the target inequality holds with $C_\star(d)=C$ and no polylogarithmic factor. The conjecture is presented as the missing bound needed to turn the KL strong data-processing implication into the desired single-round product bound; its resolution is not supplied here.