Stability of Grassmannian cluster variables under increasing relabellings

Let $T\in\operatorname{SSYT}(k,[n])$ be a tableau whose distinct entries are $a_1<\ldots<a_r$. Let $n'\ge n$ and let $f:\{a_1,\ldots,a_r\}\to[n']$ satisfy $f(a_1)<\ldots<f(a_r)$; write $f(T)$ for the tableau obtained by applying $f$ to the entries of $T$. Relabelling stability conjecture. The tableau $T$ is a cluster variable in $\mathbb{C}[\operatorname{Gr}(k,n)]$ if and only if $f(T)$ is a cluster variable in $\mathbb{C}[\operatorname{Gr}(k,n')]$. This conjecture predicts that the cluster-variable property depends only on the order pattern of the tableau entries, and its resolution is not specified in the source.