Weak differential subordination estimate for finite-dimensional Banach spaces

Let $X$ be a finite dimensional Banach space and let $p\in(1,\infty)$. For continuous martingales $M,N:\mathbb R_+\times\Omega\to X$, say that $N$ is weakly differentially subordinated to $M$ when the corresponding weak differential subordination condition holds. Let $\beta_{p,X}$ denote the constant appearing in the paper's martingale estimate.

Weak differential subordination conjecture. There exists $C_p\geq 1$ such that, for every pair of continuous martingales $M,N:\mathbb R_+\times\Omega\to X$ with $N$ weakly differentially subordinated to $M$, one has, for every $t\geq0$,

$$(\mathbb E\|N_t\|^p)^{\frac1p}\leq C_p\beta_{p,X}(\mathbb E\|M_t\|^p)^{\frac1p}.$$

By the cited theorem, this estimate would imply the paper's main estimate for Fourier multipliers. The conjecture is established in some special cases, including finite-dimensional Hilbert spaces, but is open for general finite-dimensional Banach spaces.