The edge-pattern convex-combination conjecture

Let $P$ be a $d$-regular edge-rooted pattern of depth $r$ and girth $2r+2$. For each edge-rooted pattern, let $\mathbf{e}(P)$ denote its constraint vector, and let $\mathcal{T}'_r(d)$ be the specified family of $d$-regular edge-rooted trees of depth $r$. The constraint $\mathbf{e}(P)$ is weaker than a convex combination of constraints from this family in the sense that there exist $T_1,\dots,T_m\in\mathcal{T}'_r(d)$ and $\lambda_1,\dots,\lambda_m\in[0,1]$ with

$$\sum_{i=1}^m\lambda_i=1$$

and, for every $\bm{\alpha}\in{\left(\mathbb{Q}^+\right)}^{r+1}$,

$$\bm{\alpha}^{\!\top}\left(\sum_{i=1}^m\lambda_i\mathbf{e}(T_i)\right)\ge 1\quad\implies\quad\bm{\alpha}^{\!\top}\mathbf{e}(P)\ge 1.$$

This would explain why the bounds obtained from edge-rooted trees remain valid at girth $2r+2$, and would generalize the cases checked for $r=2$ and $d\in\{3,4\}$.