For which of the standard large cardinal axioms $\varphi_{LC}$ and which $L$-like principles $\psi$ (e.g. GCH, $\mathrm{V}=\mathrm{HOD},$ the ground axiom, and various diamond and square principles) is it known that $\mathrm{ZFC} + \varphi_{LC} + \psi$ is equiconsistent with $\mathrm{ZFC} + \varphi_{LC}?$

A quick motivation: if one wishes to use forcing to prove the consistency of some theory $T$ relative to a large cardinal axiom below $0^{\sharp},$ say an inaccessible, one might as well start from a model of that large cardinal plus $V=L$ for a clearer picture of our ground universe. For large cardinals above $0^{\sharp}$ but still within the grasp of current inner model theory (roughly, up to a Woodin limit of Woodins), we can replace $V=L$ with a more general inner model axiom and still get most of its nice properties. For higher larger cardinals, we can't simply declare the universe is a canonical inner model, but I have heard that many of the nice features of inner models are known to be replicable via class forcing, so I'm interested to learn the state of knowledge regarding such equiconsistencies.