The rectangular Delta conjecture for decorated rectangular paths

Let $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, and let $p_{m,n}$ be the symmetric function associated with rectangular paths. Let $\Theta_{e_k}$ be the Theta operator indexed by $e_k$. For $\pi\in\mathsf{LRP}(m+k,n+k)^{\ast k}$, let $\mathsf{area}(\pi)$ denote its area statistic and $x^\pi$ its monomial weight. Write $[r]_q$ for the $q$-integer. Rectangular Delta conjecture for paths. One has

$$\left.\frac{[m+k]_q}{[d]_q}\Theta_{e_k}p_{m,n}\right\rvert_{q=1}=\sum_{\pi\in\mathsf{LRP}(m+k,n+k)^{\ast k}}t^{\mathsf{area}(\pi)}x^\pi.$$

Together with the preceding rectangular Delta formula, this gives a univariate rectangular extension of the Delta and Delta-square phenomena. The source describes these conjectures as leading to a natural open problem and does not provide a general proof.