The entropic Marton-type conjecture for random variables over $\mathbb{F}_2^m$

Let $c_1>0$. For a random variable $X$ valued in $\mathbb{F}_2^m$, let $X'$ be an independent copy of $X$, and let $H$ denote Shannon entropy. For a subspace $G\leq\mathbb{F}_2^m$, let $U_G$ be the uniform random variable on $G$, independent of $X$.

Entropic Marton-type conjecture. There exists $c_2=\exp(O(1/c_1))$ such that, for every $m\in\mathbb{Z}_+$ and every such $X$, there is a subspace $G$ with

$$\dim G\leq c_2H(X)$$

and

$$H(U_G+X)-H(U_G)\leq (1+c_1)\bigl(H(X'+X)-H(X')\bigr).$$

The conjecture is intended to strengthen the paper's rate from $2^{-\Omega(\sqrt m)}$ to $2^{-\Omega(\sqrt m\log m)}$, which would yield vanishing block-error probability up to Shannon capacity via the bit-to-block results cited by the authors. The statement is a conjectural entropic covering assertion related to the polynomial Freiman--Ruzsa and Marton-type phenomena; its resolution status is not specified in the supplied text.