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}