
The case $m=k=0$, i.e. the Shuffle conjecture, has been proved by Carlsson and Mellit in \cite{Carlsson-Mellit-ShuffleConj-2015}. Several special cases have also been proved in the original work of Haglund, Remmel and Wilson \cite{Haglund-Remmel-Wilson-2015}.

We limit ourselves to summarize in the following table some of the most general special cases that have been proved more recently.

\begin{center}
\begin{tabular}{cc}
	Conditions & Reference \\
	\hline 
	$m=0$ and $q=0$ & \cite{Garsia-Haglund-Remmel-Yoo-2017} \\
	$m=0$ and $q=1$ & \cite{Romero-Deltaq1-2017} \\
	$m=0$ and $\<\cdot , h_{n-d}h_d\>$ & \cite{DAdderio-Iraci-polyominoes-2017} \\
	$\<\cdot , e_{n-d}h_d\>$ & Section~\ref{ssec:oor_delta} \\
	$t=0$ or $q=0$ & Section~\ref{ssec:oor_delta}
\end{tabular}
\end{center}

We should mention here also the proof of the so called \emph{$4$-variable Catalan conjecture} due to Zabrocki \cite{Zabrocki-4Catalan-2016}, which, combined with the results in \cite{DAdderio-VandenWyngaerd-2017}, gives a ``compositional'' refinement of the case $m=0$ and $\<\cdot , e_{n-d}h_d\>$.

For more results on the Delta conjecture, i.e. the case $m=0$ see \cite{Haglund-Rhoades-Shimozono-arxiv,Haglund-Xin_Lecture-Notes,Haglund-Rhoades-Shimozono-Advances,Qiu-Remmel-Sergel-Xin-2017,Remmel-Wilson-2015,Rhoades-2018,Wilson-article-2017,Wilson-PhD-2015}.