Summary: For Banach spaces \(X\) and \(Y\) and a bounded linear operator \(T:X \rightarrow Y\) we let \(\rho(T):=\inf c\) such that \[ \Biggl(AV_{\theta_l = \pm 1} \biggl\| \sum_{l=1}^\infty \theta_l \biggl( \sum_{k=\tau_{l-1}+1}^{\tau_l} h_k T x_k \biggr)\biggr\| _{L_2^Y}^2 \Biggr)^{\frac{1}{2}} \leq c \biggl\| \sum_{k=1}^\infty h_k x_k \biggr\| _{L_2^X} \] 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 \[ \sup_{\theta_k = \pm 1} \biggl\| \sum_{k=1}^\infty \theta_k h_k T x_k \biggr\| _{L_2^X} \leq c \biggl\| \sum_{k=1}^\infty h_k x_k \biggr\| _{L_2^X}. \]
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 \(\text{UMD}\)-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.