This sequent is provable.

\begin{logicproof}{2}
    p \land q & \prem \\
    q \imp \lnot \lnot r & \prem \\
    q & $\ande{2} 1$\\
    \lnot \lnot r & $\impe 2,3$ \\
    r & $\negnege 4$ \\
    p & $\ande{1} 1$ \\
    p \land r & $\andi 6,5$
\end{logicproof}