$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k and S a nonprojective simple left A-module. Suppose the projective dimension $\pd_A(S)$ of $S$ is finite. Let $n$ be a nonnegative integer such that $n+1\leq \pd_A(S)$. Then we can find 2 simple left $A$-modules $S_1, S_2$ such that $\Ext_A^{n}(S, S_1)\neq 0$, $\Ext_A^{n+1}(S, S_2)\neq 0$. My question is the following:

Can we always choose nonisomorphic simple $A$-modules $S_1, S_2$ in the above situation?