Lexicographic edge-isoperimetric conjecture for generalized expanded Sierpinski graphs

Let $S(n,m)$ be the generalized and expanded Sierpinski graph, with vertex set $V_{S(n,m)}=\left\{0,1,\ldots,m-1\right\}^{n}$. Let $Lex$ denote the lexicographic order on this vertex set, let $\Theta(S)$ be the edge-boundary of a vertex set $S$, and let $\left\lvert\Theta\right\rvert(S(n,m);\ell)$ denote the minimum edge-boundary among subsets of $V_{S(n,m)}$ of cardinality $\ell$. Lexicographic edge-isoperimetric conjecture. The graph $S(n,m)$ has nested solutions for the edge-isoperimetric problem: for every $\ell$ with $0\leq\ell\leq m^n$, the initial $\ell$-segment of $Lex$ minimizes the edge-boundary, so that

$$\left\lvert\Theta\left(Lex^{-1}\left(\left\{0,1,\ldots,\ell\right\}\right)\right)\right\rvert=\left\lvert\Theta\right\rvert(S(n,m);\ell).$$

This would establish that lexicographic initial segments solve the edge-isoperimetric problem for every cardinality, beyond the cardinalities at which the sharp lower bound is already known. The source presents the claim as an unproved extension of the preceding theorem; its status is therefore open.