Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that $\operatorname{Ext}^*_A(M,M)$ carries an $A_\infty$-structure from which one can reconstruct $\operatorname{filt}(M)$ (in fact for this one only needs the restriction of the $A_\infty$-structure on $\operatorname{Ext}^{0,1,2}(M,M)$.
It is well known that an algebra is Koszul if and only if the Ext-algebra of the simples is formal as an $A_\infty$-algebra. Of course, one can think of formality of the Ext-algebra of other (special) modules as well. In my case, this is the direct sum of all standard modules over a quasi-hereditary algebra, but this is maybe not so much of importance. One thing that this implies is that the Ext-algebra is generated by the degree $0$ and $1$ part as an $A_\infty$-algebra.
This motivates the following question:
Suppose you have that the restriction of the $A_\infty$-structure (i.e. all higher multiplications) to $\operatorname{Ext}^{0,1,2}(M,M)$ vanishes and that the restriction of the $A_\infty$-structure to the $A_\infty$-subalgebra $\operatorname{Ext}^{\geq 1}(M,M)$ vanishes. Does it follow that $\operatorname{Ext}^*(M,M)$ is formal as an $A_\infty$-algebra?
I would conjecture that this does not follow, but I was not able to find a counterexample. A minimal counterexample could have $m_3(0,1,3)\neq 0\in \operatorname{Ext}^3(M,M)$ where $0,1,3$ stands for the respective degrees.
