In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak equivalence (i.e. $A_1\to A_2$ and $B_1\to B_2$ are weak equivalences). I need to conclude $C_1\to C_2$ is a weak equivalence. I'd love it if this were true all the time, but it probably isn't. Are there any standard axioms on a model category which let me conclude this is true? For example, left properness? I can't assume any of the objects are either cofibrant or fibrant, and the levelwise weak equivalences won't be fibrations or cofibrations.

Note that reflexive coequalizers often come up when one studies model categories because they are important for building objects of interest in categories of algebras over a monad (and to prove such categories are cocomplete). I've googled around quite a bit and can't find anything saying reflexive coequalizers preserve weak equivalences, so I'm pretty sure it's not true in general (and probably you can find a counterexample just in $Ch(R)$ via $A\otimes_R B$ or doing something similar to this) but I would really love to hear expert opinions on extra hypotheses which guarantee this.