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This sequent is provable.
\begin{logicproof}{3}
\begin{subproof}
p \land q & \assum \\
p & $\ande{1} 1$ \\
q & $\ande{2} 1$ \\
\begin{subproof}
\neg p \lor \neg q & \assum \\
\begin{subproof}
\neg p & \assum \\
\bot & $\nege 2,5$
\end{subproof}
\begin{subproof}
\neg q & \assum \\
\bot & $\nege 3,7$
\end{subproof}
\bot & $\ore 4,5-6,7-8$
\end{subproof}
\neg (\neg p \lor \neg q) & $\negi 4-9$
\end{subproof}
(p \land q) \imp \neg (\neg p \lor \neg q) & $\impi 1-10$
\end{logicproof}