Lv–Zhang joint equidistribution conjecture for mesh patterns P₁₁ and P₁₂

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and for a mesh pattern $\mathrm{P}$ let $\mathrm{P}(\sigma)$ denote the number of occurrences of $\mathrm{P}$ in $\sigma$. Lv–Zhang's conjecture. For $n\geq 1$ and any $k,\ell\geq 0$,

$$|\{\sigma\in\mathfrak{S}_n: \mathrm{P}_{11}(\sigma)=k,\ \mathrm{P}_{12}(\sigma)=\ell\}|=|\{\sigma\in\mathfrak{S}_n: \mathrm{P}_{12}(\sigma)=k,\ \mathrm{P}_{11}(\sigma)=\ell\}|.$$

The paper states that it resolves all seven Lv–Zhang joint equidistribution conjectures, so this equality is solved.