This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects Suppose that we have $$ L :C\leftrightarrow D: R$$ an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.

Suppose that

1- the functor $L$ (left adjoint) takes fibrant-cofibrant objects to fibrant-cofibrant objects.

2- the functor $R$ (right adjoint) takes cofibrations to cofibrations and takes acyclic cofibrations to acyclic cofibrations.

Is it true that $L$ sends fibrant objects to fibrant objects ?

Maybe a more general question would be the following: Under which additional (non trivial) conditions the functor $L$ would takes fibrant objects to fibrant objects ?