QF-1 characterisation conjecture for higher Auslander algebras

Let $A$ be a higher Auslander algebra of even global dimension $g$. Write $D(A)$ for the dual of $A$, and let $\tau_g$ and $\tau_g^{-1}$ denote the higher Auslander–Reiten translations. The algebra $A$ is called QF-1 when every faithful projective-injective module is a generator-cogenerator. An algebra is $g$-quasi-tilted when $\operatorname{gldim} A \leq g$ and $\operatorname{pdim} M+\operatorname{idim} M\leq 2g-1$ for every indecomposable $A$-module $M$.

QF-1 characterisation conjecture. The following conditions are equivalent:

$$A\text{ is QF-1};$$ $$A\text{ is }g\text{-quasi-tilted};$$ $$\operatorname{pdim} \tau_g(D(A))\leq g-1;$$ $$\operatorname{idim} \tau_g^{-1}(A)\leq g-1.$$

This conjecture would characterise QF-1 higher Auslander algebras of even global dimension through both the $g$-quasi-tilted condition and homological bounds involving higher Auslander–Reiten translations. The paper establishes several implications and equivalent formulations, but the full equivalence remains open.