In \cite{DAdderio-Iraci-VandenWyngaerd-Delta-Square} we proved the following results.

\begin{proposition} \label{prop:t0}
	For $n\in \mathbb{N}$, $n >0$,
	\begin{equation}
	\left.\Delta_{h_m} \Delta_{e_{n-1}}'e_n\right|_{t=0} = \mathsf{PLD}_{\underline x,q,0}(m,n)^{\ast 0}.
	\end{equation}
\end{proposition}
\begin{theorem} \label{thm:SFinvoDelta}
	Given $m,n\in \mathbb{N}$, $m\geq 0$ and $n\geq 1$, we have the two identities
	\begin{align} 
	\sum_{s= 0}^{n-1}(-t)^s\Delta_{h_m}\Delta_{e_{n-s-1}}'e_n & = \left.\Delta_{h_m}\Delta_{e_{n-1}}' e_n \right|_{t=0}
	\end{align}
	and
	\begin{align}
	\sum_{s= 0}^{n-1}(-t)^s\frac{[n-s]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-s}}\omega(p_n) & = \left.\Delta_{h_m}   \Delta_{e_{n-1}}' e_n\right|_{t=0} \,\, . 
	\end{align}
\end{theorem}

The last theorem, combined with the Delta conjectures, suggested the following one.

\begin{theorem}
	Given $m,n\in \mathbb{N}$, $m\geq 0$ and $n\geq 1$, we have the two identities
	\begin{align}
	\sum_{s=0}^{n-1}(-t)^s\mathsf{PLSQ^E}_{\underline{x},q,t}(m,n)^{\ast s} & = \mathsf{PLD}_{\underline{x},q,0}(m,n)^{\ast 0} \\
	\sum_{s=0}^{n-1}(-t)^s\mathsf{PLD}_{\underline{x},q,t}(m,n)^{\ast s} & = \mathsf{PLD}_{\underline{x},q,0}(m,n)^{\ast 0}.
	\end{align}
\end{theorem}
\begin{proof}
Fix $m,n\in \mathbb{N}$, $m\geq 0$ and $n>0$. Let 
\begin{equation}
X\coloneqq\bigsqcup_{k=0}^{n-1}\mathsf{PLSQ^E}(m,n)^{\ast k},
\end{equation}
and define a map $\varphi \colon X \to X$ in the following way: if $P\in X$ has no rises, i.e. no two consecutive vertical steps, then $\varphi(P) \coloneqq P$; otherwise, consider the first rise encountered by following the path $P$ starting from its breaking point (notice that this rise will always occur before the north-east corner): if the rise is decorated/undecorated, then $\varphi(P)$ is the path obtained from $P$ by undecorating/decorating that rise. Observe that $\varphi$ is clearly an involution, whose fixed points are the paths $P\in X$ with no rises, i.e. the paths of area $0$ with no decorated rises. Notice also that $\varphi$ restricts to an involution of
\begin{equation}
Y\coloneqq\bigsqcup_{k=0}^{n-1}\mathsf{PLD}(m,n)^{\ast k}\subset X.
\end{equation}

For any $P\in X$ we define a \emph{weight} by setting
\begin{equation}
wt(P)\coloneqq(-t)^{\mathsf{dr}(P)}q^{\dinv(P)}t^{\area(P)}x^P
\end{equation}
where $\mathsf{dr}(P)$ is defined to be the number of decorated rises of $P$.

Observe that
\begin{equation}
\sum_{P \in X}wt(P) = \sum_{s=0}^{n-1}(-t)^s\mathsf{PLSQ^E}_{\underline{x},q,t}(m,n)^{\ast s}
\end{equation}
and
\begin{equation}
\sum_{P \in Y}wt(P) = \sum_{s=0}^{n-1}(-t)^s\mathsf{PLD}_{\underline{x},q,t}(m,n)^{\ast s}.
\end{equation}

Suppose that $P\in X$ is such that $\varphi(P)\neq P$. Notice that the rise occurring in the definition of $\varphi$ is always at distance $1$ from the base diagonal, so undecorating/decorating it when it is decorated/undecorated gives $\mathsf{dr}(\varphi(P))=\mathsf{dr}(P)\mp 1$, but $\area(\varphi(P))=\area(P)\pm 1$. Since the decorations of the rises do not affect the dinv, we deduce
that $wt(\varphi(P))=-wt(P)$. This shows that in the sum $\sum_{P \in X}wt(P)$ all the contributions of the $P$ that are not fixed by $\varphi$ cancel out, leaving the sum over the fixed points of $\varphi$, i.e. over the paths with no rises, as we claimed.

The same argument applies to the sum $\sum_{P \in Y}wt(P)$.
\end{proof}


These last results combined give relations among the Delta conjectures. We get immediately the following curious corollary.
\begin{corollary}
	For fixed $m,n\in \mathbb{N}$, with $m\geq 0$ and $n> 0$, the truth of the generalized Delta (square) conjectures for all values of $k$ in $\{0,1,\dots,n-1\}$ except one imply the truth of the missing case.
\end{corollary}