This sequent is provable.

\begin{logicproof}{1}
    p \lor \lnot \lnot q & \prem \\
    \lnot p \land \lnot q & \prem \\
    \begin{subproof}
        p & \assum \\
        \lnot p & $\ande{1} 2$ \\
        \bot & $\nege 3,4$ \\
        s \lor \lnot t & $\bote 5$
    \end{subproof}
    \begin{subproof}
        \lnot \lnot q & \assum \\
        \lnot q & $\ande{2} 2$ \\
        \bot & $\nege 7,8$\\
        s \lor \lnot t & $\bote 9$
    \end{subproof}
    s \lor \lnot t & $\ore 1, 3-6, 7-10$
\end{logicproof}