Is the fiberwise suspension of a Serre fibration a Serre fibration?

Question

Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma_f X = X \times [0,1] / {\sim}$, where $\sim$ identifies $(x,0)$ with $(x',0)$ whenever $\pi(x) = \pi(x')$ and also $(x,1)$ with $(x',1)$ whenever $\pi(x) = \pi(x')$.

Question: Must $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ also be a Serre fibration?

I think this is probably false for Hurewicz fibrations, and I would also be interested in counterexamples in that more general context.

Answer 1

Lemma 6 in Section 3 of

Vandembroucq, Lucile, Fibrewise suspension and Lusternik-Schnirelmann category, Topology 41, No. 6, 1239-1258 (2002). ZBL1009.55002.

(also available at https://core.ac.uk/download/pdf/82041491.pdf) seems to show that the fibrewise suspension of a Hurewicz fibration is a Hurewicz fibration. The proof uses lifting functions.

Answer 2

"In any attempt to solve those two problems, one runs into a third one that concerns a basic foundational problem in ex-space theory. Model theoretical considerations lead to the use of Serre fibrations as projections, or to the even weaker class of qf-fibrations. However, only Hurewicz fibrations are considered in most of the literature. There is good reason for that. Fiberwise smash products, suspensions, cofibers, function spaces, and other fundamental constructions in ex-space theory do not preserve Serre fibrations."

From the introduction to Chapter 8 of "Parametrized Homotopy Theory" by May-Sigurdsson. I could not find a specific counter example stated.