Tower Sealing in generic extensions of hod pairs with reflecting strong cardinals

Let $({\mathcal{P}},\Psi)$ be a hod pair such that

$${\mathcal{P}}\vDash “\text{there is a strong cardinal which reflects the class of strong cardinals and there is a proper class of Woodin cardinals}”.$$

Let $\kappa$ be the least strong cardinal which reflects the class of strong cardinals, and let $g\subseteq\operatorname{Coll}(\omega,\kappa^+)$ be generic. Tower Sealing conjecture.

$${\mathcal{P}}[g]\vDash “\forall\delta\ \text{if }\delta\text{ is Woodin, then Tower Sealing holds at }\delta”.$$

This conjecture proposes that Tower Sealing holds at every Woodin cardinal in the indicated generic extension, in the setting where strong cardinals reflect the class of strong cardinals. The preceding discussion explains that the argument in the paper does not cover this case and that a different proof is required; the conjecture is presented as a natural direction for extending the result.