I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.

Suppose we are given a (strict) pullback square

$\begin{array}{ccc}W&\to&X\\ \downarrow && \downarrow\rlap{p}\\ S&\to&T\end{array}$

of quasi-categories with $p$ a categorical fibration; i.e. a fibration in the Joyal model structure, so it is a homotopy pullback in the Joyal model structure.

Question

Are there any criteria for the square to be a homotopy pullback also in Quillen model structure?

I also appreciate criteria with additional assumption on $p$; e.g. being a left / right fibration.