Logarithmic sparse-domination conjecture for Banach function spaces

Let $T$ satisfy the full-range sparse domination bound referred to in the source. Let $X$ be a Banach function space such that $M:X\to X$ and $M:X'\to X'$ are bounded. Logarithmic sparse-domination conjecture. Then

$$\|T\|_{X\to X}\lesssim_d C_T\max\big(\|M\|_{X\to X},\|M\|_{X'\to X'}\big)\Big(1+\log\big(\max\big(\|M\|_{X\to X},\|M\|_{X'\to X'}\big)\big)\Big).$$

The conjecture seeks a logarithmic improvement over the product of the two maximal-operator norms, motivated by the Hilbert-transform lower bound mentioned in the source. Its resolution is not supplied.