This sequent is provable.

\begin{logicproof}{2}
    p \imp q \lor r & \prem \\
    \neg q & \prem \\
    \begin{subproof}
        p & \assum \\
        q \lor r & $\impe 1,3$ \\
        \begin{subproof}
            q & \assum \\
            \bot & $\nege 2,5$ \\
            r & $\bote 6$
        \end{subproof}
        \begin{subproof}
            r & \assum \\
            r & $\copying 8$
        \end{subproof}
        r & $\ore 4, 5-7, 8-9$
    \end{subproof}
    p \imp r & $\impi 3-10$
\end{logicproof}