Vector-valued extension conjecture for rearrangement operators on UMD spaces

Let $T$ be the scalar-valued rearrangement operator associated with a rearrangement of the dyadic Haar system, and let $L^p_E$ denote the Bochner space of $E$-valued functions. Assume that $T$ is an isomorphism on $L^p$ for some $p\ne 2$. Vector-valued extension conjecture. Is $T\otimes\operatorname{Id}_E$ an isomorphism on $L^p_E$ for $1<p<\infty$ and every Banach space $E$ in the UMD class? The question concerns whether scalar isomorphisms of rearrangement operators extend to all UMD-valued $L^p$ spaces; boundedness alone is insufficient, since bounded scalar rearrangements with non-bounded vector-valued extensions are known.