AFO's stability conjecture for normal bundles of general canonical curves

Let $C$ be a general non-hyperelliptic smooth projective curve of genus $g\geq 7$, canonically embedded as $C\hookrightarrow\mathbb{P}^{g-1}$, and let $\mathcal{N}_{C/\mathbb{P}^{g-1}}$ denote its normal bundle. AFO's stability conjecture. The normal bundle $\mathcal{N}_{C/\mathbb{P}^{g-1}}$ is stable. This conjecture extends the known result that the general canonical curve of genus $7$ has stable normal bundle, while contrasting with the instability known for tetragonal curves of genus at least $6$. The assertion remains open in the stated generality.