Local Torelli conjecture for osculating cones to Brill–Noether loci

Let $C$ be a general canonically embedded curve of genus $g$, and let $L\in W^1_d(C)$ be a general point with $\dim(W^1_d(C))\geq 1$. Set

$$V=\operatorname{Sing}\bigl(\mathbb{P}\mathcal{T}_{L}(W^0_d(C))\bigr).$$

Let $\pi_V:C\to C'\subseteq\mathbb{P}^1\times\mathbb{P}^{h^1(C,L)-1}$ be the projection from $V$, birational to the image of $C$. Consider the strict transforms after blowing up $V$ and the induced projection maps

$$\widetilde{\mathbb{P}(\operatorname{OC}_3(W^0_d(C),L))}\xrightarrow{\alpha}\widetilde{\mathbb{P}\mathcal{T}_{L}(W^0_d(C))}\hookrightarrow\widetilde{\mathbb{P}^{g-1}}\xrightarrow{\pi_V}\mathbb{P}^{g-1}.$$

Local Torelli conjecture. The following hold: (a) away from points of $C'$, the fibers of $\alpha$ are smooth or empty; and (b) for a smooth point $p'$ of $C'$, the corresponding point $p$ of $\widetilde C$ is the only singular point of the fiber of $\alpha$ over $p'$. The conjecture asserts that these fibers retain enough information to recover the curve from the osculating-cone construction; the source presents it as a conjecture and supplies no resolution.