Characterization of the Barnes multiple zeta-function limit

Let $t=\varphi(s)$ be a meromorphic function on $\mathbb{C}$, and let $\zeta_{q,r}(s,t,z;\boldsymbol{\omega})$ and $\zeta_r(s,z;\boldsymbol{\omega})$ denote the corresponding $q$-analogue and Barnes multiple zeta functions. Limit characterization. The formula

$$\lim_{q\uparrow 1}\zeta_{q,r}(s,\varphi(s),z;\boldsymbol{\omega})=\zeta_r(s,z;\boldsymbol{\omega}) \qquad (s\in\mathbb{C})$$

holds if and only if $\varphi(s)=s-\nu$ for some $\nu\in\mathbb{N}$. This characterizes precisely the meromorphic substitutions in the second variable for which the $q$-analogue converges to the Barnes multiple zeta function as $q$ tends to $1$; the supplied text does not indicate whether the assertion has been proved or remains open.