We call a cardinal $\theta$ worldly if $V_\theta \models \mathsf{ZFC}$. If $\kappa$ is an inaccessible cardinal, then we have an unbounded set of worldly cardinals below $\kappa$, in fact even a club set (by considering a chain of elementary submodels of $V_\kappa$).

My question is whether it is possible to destroy the worldly cardinals below an inaccessible cardinal by forcing in a mild way in the following sense: given an inaccessible $\kappa$ can there be a forcing that destroys the worldliness of all worldly cardinals below $\kappa$ while preserving the worldliness of $\kappa$ itself?

Of course some further assumptions will be necessary, since such a forcing must singularize $\kappa$, which implies that $\kappa$ is measurable in some inner model, but I am just asking whether the existence of such a forcing is consistent relative to any appropriate large cardinal assumption.