Subadditivity-plus-sigma conjecture for lexicographic edge boundaries

For $0\leq \ell_b\leq\ell_a\leq m^n$, define

$$\sigma_{n,m}(\ell_a,\ell_b)= \begin{cases} q_{n,m}(\ell_b)+q_{n,m}(\ell_a+\ell_b)-q_{n,m}(\ell_a),&\ell_a+\ell_b<m^n,\\ q_{n,m}(\ell_b)-q_{n,m}(\ell_a+\ell_b-m^n)+m-q_{n,m}(\ell_a),&\ell_a+\ell_b\geq m^n. \end{cases}$$

Here $q_{n,m}$ is the previously defined auxiliary function, $\operatorname{Lex}^{-1}(n,m;\ell)$ is the lexicographic initial $\ell$-segment in $S(n,m)$, and $|\Theta(\operatorname{Lex}^{-1}(n,m;\ell))|$ is its edge-boundary size.

Subadditivity-plus-sigma conjecture. For all $n,m,\ell_a,\ell_b\in\mathbb N$ with $m^n\geq\ell_a\geq\ell_b>0$, if $\ell_a+\ell_b\leq m^n$, then

$$|\Theta(\operatorname{Lex}^{-1}(n,m;\ell_a+\ell_b))|+\sigma_{n,m}(\ell_a,\ell_b) \leq |\Theta(\operatorname{Lex}^{-1}(n,m;\ell_a))|+|\Theta(\operatorname{Lex}^{-1}(n,m;\ell_b))|.$$

If $\ell_a+\ell_b\geq m^n$, then

$$|\Theta(\operatorname{Lex}^{-1}(n,m;\ell_a+\ell_b-m^n))|+\sigma_{n,m}(\ell_a,\ell_b) \leq |\Theta(\operatorname{Lex}^{-1}(n,m;\ell_a))|+|\Theta(\operatorname{Lex}^{-1}(n,m;\ell_b))|.$$

For $m=3$, related strengthened subadditivity inequalities had already been proved, while this statement is presented as their generalization to arbitrary $m$. It was introduced as a sufficient ingredient for proving the main lexicographic edge-isoperimetric conjecture.