The local cohomology conjecture for additive Okamoto–Painlevé pairs

Let $(S,Y)$ be a generalized rational Okamoto–Painlevé pair of non-fibered type, let $D=Y_{\mathrm{red}}$ be a normal crossing divisor with at least two irreducible components, and suppose that $D$ is of additive type. Write $N_D={\cal O}_S(D)/{\cal O}_S$. The preceding theorem gives

$$\dim H^0\left(D,\Theta_S(-\log D)\otimes N_D\right)=1.$$

Local cohomology conjecture. Under these assumptions,

$$H^1_D\left(\Theta_S(-\log D)\right)\simeq H^0\left(D,\Theta_S(-\log D)\otimes N_D\right)\simeq {\bold C}.$$

The conjecture would identify the local cohomology space governing the deformation with the one-dimensional space of sections appearing in the theorem, strengthening the known natural inclusion and the resulting lower bound on its dimension. Its resolution is not specified in the source.