Is there a 2-categorical, equivariant version of Quillen's Theorem A?

Quillen’s Theorem A says that a functor $F:C to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ has a contractible classifying space.

Quick reminder: the objects of $F/d$ are pairs $(c,gamma)$ consisting of an object $c$ in $C$ and a morphism $gamma:Fc to d$ in $D$; and morphisms $(c,gamma) to (c’,gamma’)$ in $F/d$ are given by all $f:c to c’$ in $C$ satisfying $gamma = gamma’ circ Ff$ in $D$.

Here are two natural extensions of this result:

1. If a group $Gamma$ acts on $C$ and $D$ with $F$ an $Gamma$-equivariant functor, then $F$ induces a $Gamma$-equivariant homotopy equivalence between $C$ and $D$ provided its fibers are $Gamma$-equivariantly contractible. I must confess that the only published version that I am aware of restricts to the case where $C$ and $D$ are $Gamma$-posets, due to Thevanaz and Webb (pdf here)
2. If we’re working with a (lax/oplax) 2-functor between $2$ categories, then Quillen’s Theorem A holds provided we define the lax fiber 2-categories $F/d$ correctly; I learned this from Bullejos and Cegarra’s paper (pdf, see Thm 2)

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My question is, is the union of 1 and 2 published somewhere? Which is to say, if I have a (lax/oplax) functor $F:C to D$ between $2$-categories which is equivariant with respect to the action of some group $Gamma$, and if its fiber 2-categories are $Gamma$-equivariantly contractible, can I automatically claim that $F$ induces a $Gamma$-equivariant homotopy equivalence between classifying spaces?

What I’m hoping to avoid is the scenario where the result is decipherable "only from the corresponding result for sheaves on $(infty,1)$-topoi" or something similar. For reasons both psychological and mathematical, I’d like to restrict the categorical depth to $leq 2$ if at all possible.