One can show that spectrifications of maps between $\Sigma$-cofibrant prespectra that are spacewise homotopy equivalences are homotopy equivalences of spectra. I'm interested in the following:

Does this result hold if only the source prespectrum is assumed to be $\Sigma$-cofibrant?

The reason for my interest is the following. The adjoint from prespectra to inclusion prespectra is difficult to deal with. I see that sometimes people replace a prespectrum by a thickened one, via a cylinder construction, as in Chapter X, Section 5 of Elmendorf-Kriz-Mandell-May's "Rings, modules and algebras in stable homotopy theory", to get a $\Sigma$-cofibrant prespectrum so that we have a nicer description of the spectrification. This is done, for example, when defining the topological Hochschild spectrum in Hesselholt-Madsen's "On the $K$-theory of finite algebras over Witt vectors of perfect fields". Say the original prespectrum is $t$, the cylinder construction yielding a $\Sigma$-cofibrant prespectrum is denoted by $K$ and spectrification is denoted by $L$. We have a map of prespectra $Kt \to t$ which is a spacewise homotopy equivalence. I would hope that $LKt$ and $Lt$ are related in a nice way so as to say that the intermediate thickening didn't essentially change anything. The "nice way" I'm hoping for is that one can deduce that the induced map $LKt \to Lt$ is a homotopy equivalence (or perhaps weak homotopy equivalence).