The self-duality conjecture for powers of canonical modules of analytic spread one

Let $R$ be a standard graded algebra over a field. Assume that $R$ is reduced and equidimensional, and let $\omega_R$ denote its canonical module, generated in a single degree. The canonical module has analytic spread 1 when its corresponding fractional ideal has analytic spread one.

Canonical-module self-duality conjecture. If $\omega_R$ has analytic spread $1$, then some power of $\omega_R$, regarded as a fractional ideal, is $\omega_R$-self-dual up to a shift.

This conjecture is suggested by further examples and is proposed independently of the residual-intersection construction. The stated source provides no resolution, so the existence of such a self-dual power remains open in this generality.