Lusin cotype characterization of martingale cotype for Schrödinger semigroups

Let $\{T_t\}_{t>0}$ be the heat semigroup generated by a Schrödinger operator as above, and let $\{P_t\}_{t>0}$ be the Poisson semigroup subordinated to $\{T_t\}_{t>0}$. Let $X$ be a Banach space, and let $q$ be the cotype exponent in question. Lusin cotype conjecture. If $X$ is of Lusin cotype $q$ relative to $\{P_t\}_{t>0}$, then $X$ is of martingale cotype $q$. The same implication should hold for the heat semigroup $\{T_t\}_{t>0}$ itself when the underlying differential operator $L$ is uniformly elliptic. This extends the known characterization for the Laplacian on $\mathbb{R}^d$ with $d\geq 3$ and a potential satisfying a reverse Hölder inequality; the general Schrödinger-semigroup case remains conjectural.