A cardinal $\kappa$ is a Vopěnka cardinal if Vopěnka's principle holds in $V_\kappa$.

Suppose that $\kappa$ and $\lambda$ are both Vopěnka cardinals with $\lambda > \kappa$. Must it be the case that $V_\lambda \vDash$ '$\kappa$ is a Vopěnka cardinal'?

(Understand that the predicate "… is a Vopěnka cardinal" after the turnstile as standing for a schema, since the informal claim is not directly expressible in $\mathsf{ZFC}$.)