The Delta square conjecture

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by proving the specialization $m=q=0$, reducing it to the same case of the Delta conjecture, and the Schr\"{o}der case, i.e. the case $\langle \cdot ,e_{n-d}h_d\rangle$. The latter provides a broad generalization of the $q,t$-square theorem of Can and Loehr (Can, Loehr 2006). We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed $m$ and $n$. Finally, in the appendix, we give a new proof of the Delta conjecture at $q=0$.