Clarke–Larson–Vogt's conjecture on stability of normal bundles of BN-curves

A BN-triple is a triple $(g,r,d)$ specifying a Brill–Noether curve of genus $g$, degree $d$, and dimension $r$. For a general BN-curve $X\subset\mathbb{P}^r$ of type $(g,r,d)$, consider its normal bundle $N_{X/\mathbb{P}^r}$. Clarke–Larson–Vogt's conjecture. All but finitely many BN-triples $(g,r,d)$ with $g\geq2$ are stable, meaning that $N_{X/\mathbb{P}^r}$ is stable for a general BN-curve of that type. The conjecture is motivated by stability results in low-dimensional cases and leaves open the possibility of finitely many exceptional BN-triples.