This sequent is provable.

\begin{logicproof}{2}
    (p \imp (q \imp r)) \land (q \imp (r \imp s)) & \prem \\
    \begin{subproof}
        p & \assum \\
        p \imp (q \imp r) & $\ande{1} 1$ \\
        q \imp (r \imp s) & $\ande{2} 1$ \\
        q \imp r & $\impe 2,3$ \\
        \begin{subproof}
            q & \assum \\
            r & $\impe 5,6$\\
            r \imp s & $\impe 4,6$\\
            s & $\impe 7,8$
        \end{subproof}
        q \imp s & $\impi 6-9$
    \end{subproof}
    p \imp (q \imp s) & $\impi 2-10$
\end{logicproof}