Generalized edge-isoperimetric conjecture for Sierpinski graphs with exterior boundary conditions

Let $I=\left\{0,1,\ldots,s-1\right\}$, $J=\left\{s,s+1,\ldots,s+t-1\right\}$, and $K=\left\{s+t,s+t+1,\ldots,m-1\right\}$, where $s,t\geq0$ and $s+t\leq m$. Let $S_{s,t}(n,m)$ be the graph $S(n,m)$ with exterior edges attached to the corner vertices indexed by $I\cup K$. In computing $\left\lvert\Theta_{s,t}(S)\right\rvert$, treat the exterior endpoint corresponding to $i\in I$ as belonging to $S$, and the one corresponding to $i\in K$ as belonging to the complement; vertices indexed by $J$ remain corner vertices without exterior edges. Let $Lex$ be the lexicographic order and let $\left\lvert\Theta_{s,t}\right\rvert(S_{s,t}(n,m);\ell)$ be the minimum boundary size among subsets of cardinality $\ell$. Generalized exterior-boundary conjecture. For every $\ell$ with $0\leq\ell\leq m^n$,

$$\left\lvert\Theta_{s,t}\right\rvert\left(S_{s,t}(n,m);\ell\right)=\left\lvert\Theta_{s,t}\left(Lex^{-1}(\ell)\right)\right\rvert.$$

This extends the proposed lexicographic optimality from the ordinary generalized expanded Sierpinski graph to graphs with prescribed exterior boundary conditions. The source gives no proof or resolution, so the conjecture remains open.