Gonericity conjecture for canonical curves

Let $C$ be an irreducible, smooth canonical curve of genus $g$ and linear colength $\ell$. A curve is goneric when its gonality is $\ell+2$ and

$$\beta_{g-2-\ell,g-1-\ell}=g-2-\ell.$$

Gonericity conjecture. In characteristic zero, $C$ is $\ell+2$-goneric if and only if

$$\beta_{g-2-\ell,g-1-\ell}=g-2-\ell,$$

unless $(g,\ell)=(6,1)$ and $C$ is isomorphic to a smooth plane quintic.

This conjecture proposes that, apart from the stated plane-quintic exception, the relevant Betti-number equality characterizes gonericity. The source gives no resolution status.