A counterexample concerning the relation between decoupling constants and $\operatorname{UMD}$-constants

For Banach spaces $X$ and $Y$ and a bounded linear operator
$T:X \rightarrow Y$ we let $\rho(T):=\inf c$ such that

for all finitely supported $(x_k)_{k=1}^\infty \subset X$ and all $0 = \tau _0 < \tau _1 < \cdots$, where $(h_k)_{k=1}^\infty \subset L_1[0,1)$ is the sequence of Haar functions. We construct an operator $T:X \rightarrow X$, where $X$ is superreflexive and of type 2, with $\rho(T)<\infty$ such that there is no constant $c>0$ with

In particular it turns out that the decoupling constants $\rho(I_X)$, where $I_X$ is the identity of a Banach space $X$, fail to be equivalent up to absolute multiplicative constants to the usual $\operatorname{UMD}$-constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.