Let $$ \begin{array}{ccc} A & \to & B \cr\downarrow&&\downarrow \cr A'& \to &B' \end{array} $$ be a pullback square in the category of all topological spaces (not just in a convenient one) and the arrow $B\rightarrow B'$ a Dold fibration.

Is the square always a homotopy pullback square

- in the Quillen model structure?
- in the Strom model structure?

In less abstract terms, is the canonical map from the pullback of the Hurewicz fibration associated to $B\rightarrow B'$ with $A'\rightarrow B'$ to $A$ a

- weak homotopy equivalence?
- homotopy equivalence?

Note that a Dold fibration may not be a Serre fibration.

isa Dold fibration (see mathoverflow.net/questions/20442/…) $\endgroup$