I'm wondering about the theoretical placement of quasifibrations.

One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback square involving (one) weak fibration is a homotopy pullback square.

Is the corresponding result true for quasifibrations in the Serre-Quillen context? That is, suppose $E\to B$ is a quasifibration, and the square $$ \begin{array}{ccc} P & \to & E \cr\downarrow&pb&\downarrow \cr X& \to &B \end{array} $$ is a categorical pullback. Then is it a homotopy pullback in the Quillen-Serre model structure?