Polynomial dimension conjecture for multiplicatively invariant sets

Fix $n,m\in\mathbb{N}$ and let

$$\Delta(A,B)=\{a^n+b^m\mid a\in A,b\in B\}$$

for $A,B\subseteq\mathbb{N}_0$. Let $r,s\in\mathbb{N}$ be multiplicatively independent, and let $A,B\subseteq\mathbb{N}_0$ be $\times r$- and $\times s$-invariant subsets, respectively. Polynomial dimension conjecture. One has

$$\dim_{\operatorname{M}}\Delta(A,B)=\min\left(\frac{1}{n}\dim_{\operatorname{M}} A+\frac{1}{m}\dim_{\operatorname{M}} B,1\right).$$

This is a special case of a natural polynomial extension of the dimension formula for sumsets of multiplicatively invariant sets. The conjecture was recently proved independently by Shmerkin and Wu.