Recall that a Banach space $B$ has Unconditional Martingale Difference (UMD-$p$) if there is a constant $C_p$ such that for every $B$-valued martingale difference sequences $(d_n)_n$ and choice of $\pm 1$ signs $(\varepsilon_n)_n$ we have $$\bigg(\textbf{E} \Big\|\sum_n \varepsilon_n d_n\Big\|^p\bigg)^{1/p} \le ~C_p \bigg(\textbf{E} \Big\|\sum_n d_n\Big\|^p\bigg)^{1/p}.$$ Let's call the UMD-p constant of the space the smallest $C_p$ such that this inequality holds.
Question: Is it known how the UMD constant behaves under interpolation (complex, real, etc.)? Ideally I'm looking for something like what complex interpolation gives for the moduli of uniform convexity and smoothness, see for example https://arxiv.org/abs/1611.08861
