Let $A$ be finite dimensional connected algebra. A simple module $S$ is called $n$-regular in case $pd(S)=n$, $Ext_A^i(S,A)=0$ for $i=0,1,...,n-1$ and $Ext_A^n(S,A)$ being a simple $A$-left module. See for example https://www.sciencedirect.com/science/article/pii/S0001870818302809 for the relvance of 2-regular simple modules. We can assume $A$ is a quiver algebra and then being $n$-regular simply means that the injective envelope $I(S)$ of $S$ (having projective dimension $n$) occurs uniquely in the minimal injective coresolution $(I_i)$ of $A$ as a summand of $I_n$.

Questions:

In case every simple module of projective dimension $n$ (and there exists at least one such simple module) is $n$-regular, does $A$ have global dimension $n$? (In case this is false, is it true when assuming $A$ hsa finite global dimension?)

When every right simple module of projective dimension $n$ is $n$-regular (and there exists at least one such simple module) is the same true for every left simple modules of projective dimension $n$?

edit: Question 1 and 2 were shown to be wrong by the answer of Erik D except the question in the brackets. For clarity I state the remaining open question here again (I also add some bonus questions, that I try to answer myself with the computer).

- In case every simple module of projective dimension $n$ (and there exists at least one such simple module) is $n$-regular, does $A$ have global dimension $n$ in case $A$ has finite global dimension? (what if we assume that this holds for simple left and right modules?)

(Bonus question: In case every simple (left and right, or maybe just onesided is enough?) module of projective dimension $n$ (and there exists at least one such simple module) is $n$-regular, does $A$ have finitistic dimension $n$?)