The discrete-family conjecture for simply connected Gorenstein differential graded algebras

Let $A$ be a simply connected Gorenstein differential graded algebra of finite type. Write $\operatorname{H}^eA$ and $\operatorname{H}^*A$ for its cohomology in degree $e$ and in all degrees, respectively, let $\mathbf{D}^c(A)$ be the compact derived category of $A$, and let $f$ be the additive function on its indecomposable objects. Assume

$$\operatorname{dim}_k\operatorname{H}^eA\leq 1\quad\text{for all }e\in\mathbb Z,$$

and $\operatorname{dim}_k\operatorname{H}^*A=3$. Discrete-family conjecture. For each $n\in\mathbb N$, there are only finitely many non-isomorphic indecomposable objects in $\mathbf{D}^c(A)$ having value $n$ under $f$. In particular, the Auslander–Reiten quiver of $\mathbf{D}^c(A)$ consists only of countably many components. This would establish that the case $\operatorname{dim}_k\operatorname{H}^*A=3$ is genuinely discrete, rather than admitting parameter families of Auslander–Reiten components; the source presents it as an open question.