Say that an inner model $M$ of $V$ is *generically saturated* if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses cardinals.

What is the consistency strength of "$L$ is generically saturated"?

If the answer is $0^\#$ exists, is this sort of a general answer for relative constructibility (i.e. $L[A]$ is generically saturated if and only if $A^\#$ exists)? If the answer is negative, what can we conclude from this principle?

Note, Mohammad Golshani remarks (also this), that assuming $0^\#$ exists for every $\kappa$, $\operatorname{Add}(\kappa,1)^L$ collapses $\kappa$ to $\omega$ (in particular, if $\kappa$ is countable in $V$, it is just a Cohen forcing).

So in the presence of $0^\#$ at least we know that a lot of the forcings in $L$ do collapse cardinals, even if they do not admit generics in $V$ (e.g., $\operatorname{Col}(\omega,\omega_1^V)$ cannot admit a generic, although it *does* collapse cardinals).

(The idea here is to marry Foreman's maximality principle that states that every forcing adds a real or collapses cardinals, with inner model hypothesis-like ideas.)