Is the Hurewicz model category left proper?

A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak equivalences are homotopy equivalences and cofibrations are defined in terms of a lifting property for Hurewicz fibrations. Is this model structure left proper?

• Isn't it true that having all objects cofibrant implies left properness? – David White May 5 '19 at 2:52

The answer is yes. I say this because I recall that the way you form homotopy pushout of a `prepushout' diagram $$C\gets A\to B$$ in the Hurewicz structure is
1. map a diagram $$\bar C \gets \bar A \to \bar B$$ in which both arrows are (Hurewicz) cofibrations into the given one by a pointwise homotopy equivalence
If $$C\gets A$$ is a homotopy equivalence, we can take the replacement to be $$\bar A \xleftarrow{id} \bar A \to \bar B$$.