Invariance of Thom spectra

Question

Suppose $R$ is a commutative ring spectrum, and let $f,g: X \to BGL_1(R)$ be $E_1$-maps. Then their Thom spectra are $E_1$ ring spectra. If $f$ and $g$ are homotopic via $E_1$-maps, does it follow that their Thom spectra are equivalent as $E_1$ ring spectra? (And not, for example, just as ring spectra up to homotopy?)

Answer 1

Yes this is true. This follows implicitly from the fact that the $E_1$-structure on $Mf$ can be identified with the canonical $E_1$-structure on $\mathrm{colim}_Xf$ (see for example here) and that is invariant under equivalences of $E_1$-map.