This sequent is provable.

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