This sequent is provable.

\begin{logicproof}{2}
    \lnot \lnot \lnot p \land q & \prem \\
    \lnot \lnot r & \prem \\
    \lnot \lnot \lnot p & $\ande{1}1$ \\
    q & $\ande{2}1$ \\
    r & $\negnege2$ \\
    \lnot p & $\negnege3$ \\
    \lnot \lnot q & $\negnegi4$ \\
    r \land \lnot p & $\andi 5,6$ \\
    r \land \lnot p \land \lnot \lnot q & $\andi 8,7$
\end{logicproof}