Geometric characterization conjecture for goneric curves

Let $C$ be a non-hyperelliptic curve over a field of characteristic zero, and let $W^1_d(C)$ denote the Brill–Noether locus of degree-$d$ line bundles with at least two independent sections. A curve is $d$-gonерic if it is $d$-gonерic in the sense of the paper.

Geometric characterization conjecture. The curve $C$ is $d$-gonерic if and only if

$$W^1_d(C)$$

is a single reduced point, and this point is the unique line bundle of degree at most $g-1$ that computes the Clifford index of $C$.

The claim gives a geometric characterization of gonericity through the Brill–Noether locus and Clifford-index-computing line bundles. The source gives no resolution status.