$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$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?



Can we always choose 2 nonisomorphic simple modules to satisfy the following nonvanishing extension conditions?
Let $A$ be the path algebra of the quiver with three vertices $1$, $2$, $3$ with arrows $a:1\to 2$, $b:2\to 3$, $c:3\to 1$, modulo relations $bca=0$, $cabc=0$, so the indecomposable projectives are uniserial modules
$$P\_1=\begin{matrix}S\_1\\S\_2\\S\_3\\S\_1\end{matrix},\quad\quad\quad
P\_2=\begin{matrix}S\_2\\S\_3\\S\_1\end{matrix},\quad\quad\quad
P\_3=\begin{matrix}S\_3\\S\_1\\S\_2\\S\_3\end{matrix}$$


Then $S\_3$ has a finite minimal projective resolution
$$0\rightarrow P\_2\rightarrow P\_1\rightarrow P\_1\rightarrow P\_3\rightarrow S\_3\rightarrow0,$$


and so $\operatorname{pd}\_A(S\_3)=3$, and both $\operatorname{Ext}^1\_A(S\_3,S\_i)$ and $\operatorname{Ext}^2\_A(S\_3,S\_i)$ are nonzero only for $i=1$.


