Existence of large half-canonical loci with injective Gaussian maps

Let $\mathcal{S}_g$ denote the moduli space of smooth genus-$g$ spin curves, and let $\mathcal{S}_g^r$ denote the locus of spin curves whose theta-characteristic has at least $r+1$ sections. For any $r\geq 3$ and $g\geq {r+2\choose 2}$, the large half-canonical locus conjecture. there exists a component of $\mathcal{S}_g^r$ of codimension ${r+1\choose 2}$ inside $\mathcal{S}_g$. This predicts a uniform construction of half-canonical curves in projective space with large genus and injective Gaussian maps, extending the examples preceding the conjecture; the supplied text gives no resolution, so the claim remains open.