When does the Ext-quiver of a finite-dimensional algebra have no oriented cycles, or what structure does it have?

Question

Let $H$ be a finite-dimensional algebra and $S(1), \dots, S(n)$ be the simple modules of $H$. Let $Q$ be the Ext-quiver of $H$, i.e. $Q$ has as vertices the simple modules $S(1), \dots, S(n)$ and an arrow $S(i) \to S(j)$ provided Ext$^1_H(S(i), S(j)) \neq 0$.

Question: What is known or conjectured about the question when there cannot be an oriented cycle in the Ext-quiver? The no loop conjecture (which is, as far as I know, only proven for algebraically closed fields, see for example A proof of the strong no loop conjecture by Igusa, Liu and Paquette) says that in finite global dimension there cannot be a loop in the Ext-quiver. Is there an analogous statement or conjecture for oriented cycles? I'm also interested in references, if you know any.

Added later: As pointed out by Stephen, there cannot be a general analogous statement. Nevertheless, I'm additionally interested in anything you know about the structure of the Ext-quiver whenever $H$ has finite global dimension, whether it is related to cycles or not.

Answer 1

Apart from the criterion you already mentioned that by the no-loop conjecture there are no loops is no further restriction on the $\operatorname{Ext}$-quiver.

In fact, for every quiver $Q$ without loops, there exists an admissible ideal $I$ such that $\operatorname{gldim} kQ/I\leq 2$, see e.g. [Happel, Zacharia: Algebras of finite global dimension, Theorem 3.1], [Poettering: Quivers without loops admit global dimension 2], [Dlab-Ringel: The dimension of a quasi-hereditary algebra], [Dlab-Ringel: Filtrations of right ideals related to projectivity of left ideals].

Answer 2

For a highest weight category $C$ with duality, and any pair of irreducible objects $L_1,L_2$, there is an isomorphism

$$\mathrm{Ext}^1_C(L_1,L_2) \cong \mathrm{Ext}^1_C(L_2,L_1)$$

This happens for instance for blocks of category $\mathcal{O}$ for Lie algebras and Cherednik algebras. These examples are all finite global dimension. So I suppose that I'd say there is no hope of something analogous. You might ask for a necessary or sufficient condition for no oriented cycles.