Morrison's stability conjecture for Hilbert points of bicanonical curves

Let $C$ be a smooth bicanonical curve of genus $g\geq 3$, and let $m^{th}$ Hilbert point denote the Hilbert point associated to its bicanonical embedding at level $m$. Morrison's conjecture. The $m^{th}$ Hilbert point of $C$ is stable whenever $(g,m)\neq (3,2)$. This conjecture concerns finite Hilbert stability for smooth bicanonical curves; the source presents it as the original motivation for studying low-degree Hilbert points, but the supplied text gives no resolution.