The consistency-strength conjecture for $d4a2\mathfrak{Sp}^* = \omega_1$

Let $\mathfrak{Sp}^*$ denote the spectrum introduced in the surrounding discussion, and let a cardinal $\kappa$ be virtually strong if for every $\gamma \geq \kappa$ and every $\delta \geq \kappa$, there is a transitive model $M$ containing $V_\delta$ such that, in the generic extension by $\operatorname{Col}(\omega,V_\gamma)$, there is an elementary embedding $j\colon V_\gamma\to M$ with critical point $\kappa$ and $j(\kappa)\geq\delta$. The consistency strength of

$$\mathfrak{Sp}^* = \omega_1$$

is exactly that of a virtually strong cardinal. The preceding theorem shows that a virtually strong cardinal suffices for $\mathfrak{Sp}^* = \omega_1$ in the relevant generic extension; the converse consistency-strength lower bound is the asserted open direction.