UMD openness conjecture

Let $X$ be an $r$-intermediate UMD space for some $r\in(2,\infty)$. Equivalently, there exist a UMD space $Y$ and a Hilbert space $H$ forming a compatible couple such that $X$ is isomorphic to $[Y,H]_{2/r}$. UMD openness conjecture. There exists $\tilde r<r$ such that $X$ is $\tilde r$-intermediate UMD. Equivalently, the set of exponents $r\in[2,\infty)$ for which $X$ is $r$-intermediate UMD is relatively open in $[2,\infty)$. This conjecture is presented as an extension of the intermediate UMD conjecture and would imply the endpoint conjecture above.