Can a Vopenka cardinal be supercompact?

Question

Can a Vopenka cardinal be supercompact?

I asked a weaker question on here before. Unfortunately, I don't know enough set theory to see whether the positive answer there generalizes to a positive answer here.

Answer 1

If $\kappa$ is almost huge with target $\lambda,$ then $V_{\lambda}$ thinks that $\kappa$ is a supercompact Vopenka cardinal.

I'll take for granted the standard facts about almost huge cardinals listed here: https://neugierde.github.io/cantors-attic/Huge

In particular, $\kappa$ is Vopenka (in $V$), and this is just an assertion about $V_{\kappa+1}$ so $V_{\lambda}$ also sees that $\kappa$ is Vopenka. Also $\kappa$ is $\theta$-supercompact for every $\theta<\lambda,$ i.e. every $\mathcal{P}_{\kappa}(\theta)$ has a normal fine measure. Again these facts are all absolute down to $V_{\lambda},$ so $V_{\lambda}$ thinks that $\kappa$ is supercompact.

Also by a standard elementary embedding argument, these facts about $\kappa$ and $\lambda$ imply that there are stationarily many $\alpha<\kappa$ such that $V_{\kappa}$ believes $\alpha$ to be a supercompact Vopenka cardinal.