QF-1 implication conjecture for higher Auslander algebras

Let $A$ be a higher Auslander algebra of even global dimension $g$, and let $D(A)$ denote the dual of $A$. Let $\tau_g$ be the higher Auslander–Reiten translation. The property QF-1 is the condition that every faithful projective-injective module is a generator-cogenerator.

QF-1 implication conjecture. If

$$\operatorname{pdim} \tau_g(D(A))\leq g-1,$$

then $A$ is QF-1.

This is the shortened form of the paper's broader characterisation conjecture, obtained from results proved earlier in the paper. The implication itself remains open according to the supplied source information.