This sequent is not provable.

Model $\mathcal{M}$:\\
\begin{minipage}{0.2\textwidth}
  \begin{align*}
  \mathcal{A} =& \; \{ 0,1 \}\\
  S^\mathcal{M} =& \; \{ (0,1),(1,1) \}\\
  T^\mathcal{M} =& \; \{ (0,1),(1,1) \}
  \end{align*}
\end{minipage}\\

$\mathcal{M} \models \exists a \forall b \; (S(b,a) \land T(b,a))$\\
$\mathcal{M} \nmodels \forall b \forall a \; (S(b,a) \land T(b,a))$