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I would like to know whether the reduced suspension of a Hurewicz cofibration of pointed spaces (it is a Hurewicz cofibration when considered as a map of unbased spaces) is an acyclic Hurewicz cofibration.

I think that this is wrong, but I have no counterexample so far (it will probably involve a degenerately based space).

NB: By space I mean compactly generated weak Hausdorff ones, but all perspectives could be interesting.

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Surprisingly enough, this is true. This follows immediately by the definition of a monoidal model category, and the fact that the category of pointed spaces under smash product and the Hurewicz model strucutre induced form the one on unbased spaces is monoidal (see More Concise Algebraic Topology for instance).

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