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This sequent is provable.
\begin{logicproof}{1}
(p \imp q) \land (q \imp r) & \prem\\
\neg r \lor q & \prem\\
p \imp q & $\ande 1$ \\
q \imp r & $\ande 1$ \\
\begin{subproof}
\neg r & \assum\\
\neg q & $\MT 4, 5$ \\
\neg p & $\MT 3, 6$ \\
\neg p \lor r & $\ori 7$
\end{subproof}
\begin{subproof}
q & \assum \\
r & $\impe 9, 4$ \\
\neg p \lor r & $\ori 10$
\end{subproof}
\neg p \lor r & $\ore 2, 5-8, 9-10$
\end{logicproof}