Decorated Sierpinski graph conjecture for the edge-isoperimetric profile

Let $S_{s,t}(n,m)$ be the decorated Sierpinski graph obtained from $S(n,m)$ by attaching exterior edges to corner vertices indexed by $I\cup K$, where $I=\{0,1,\ldots,s-1\}$, $J=\{s,s+1,\ldots,s+t-1\}$, and $K=\{s+t,s+t+1,\ldots,m-1\}$, with $s,t\in\mathbb N$ and $s+t\leq m$. Let $\Theta_{s,t}(S)$ denote the corresponding boundary, and let $\operatorname{Lex}^{-1}(n,m;\ell)=\operatorname{Lex}^{-1}(\{1,2,\ldots,\ell\})$ in $S_{s,t}(n,m)$. Write $|\Theta_{s,t}|(S_{s,t}(n,m);\ell)$ for the minimum boundary size among $\ell$-vertex sets.

Decorated Sierpinski graph conjecture. For every $\ell$ with $0\leq\ell\leq m^n$,

$$|\Theta_{s,t}|(S_{s,t}(n,m);\ell)=|\Theta_{s,t}(\operatorname{Lex}^{-1}(n,m;\ell))|.$$

This conjecture was stated in the cited earlier work as a decorated analogue of the lexicographic edge-isoperimetric assertion. The supplied text gives no evidence that it has been resolved.