The UMD$^-_{\rm PW}$ conjecture for Musielak–Orlicz spaces

Let $T$ be the underlying measure space and let $\Phi:T\times [0,\infty)\to [0,\infty)$ be a Young function satisfying the $\Delta_2$ condition. The Musielak–Orlicz space $L^{\Phi}(T)$ is defined by $\Phi$. A Banach space $X$ is UMD$^-_{\rm PW}$ if there exist $p\in[1,\infty)$ and $C\geq 0$ such that every Paley–Walsh martingale $f$ satisfies

$$\|f\|_{L^p(\Omega;X)}\leq \|\varepsilon'*f\|_{L^p(\Omega\times\Omega';X)}.$$

The UMD$^-_{\rm PW}$ conjecture. If $\Phi\in\Delta_2$, then $L^{\Phi}(T)$ is UMD$^-_{\rm PW}$.

This would extend the known UMD results to the broader randomized class of UMD$^-_{\rm PW}$ spaces. The conjecture is open, including when $\Phi$ does not depend on $T$.