The local cohomology conjecture for additive generalized Okamoto–Painlevé pairs

Let $(S,Y)$ be a generalized rational Okamoto–Painlevé pair satisfying the conditions above, let $D=Y_{\mathrm{red}}$ be of additive type, and set

$$N_D={\mathcal O}_S(D)/{\mathcal O}_S.$$

The theorem above gives $\dim H^0(D,\Theta_S(-\log D)\otimes N_D)=1$ and a natural inclusion of this space into $H^1_D(\Theta_S(-\log D))$.

Local cohomology conjecture. Under the same notation and assumptions,

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

This proposes that the local cohomology group is exactly the one-dimensional space supplied by the theorem, strengthening the known lower bound and relating the deformation-theoretic local cohomology to the time direction of the Painlevé equation associated with $(S,Y)$.