It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter added by the first poset. That is, Mathias forcing is the two-step iteration of a countably closed poset and a $\sigma$-centered poset. In a similar way, Laver forcing completely embeds into a closed $\ast$ centered poset. It’s natural to wonder whether their cousin Miller forcing (aka rational perfect set forcing) has the same property. While Laver and Mathias forcings have natural ccc versions guided by an ultrafilter, it’s not clear (to me) how to guide Miller forcing by an ultrafilter. Here is my question:

Does Miller forcing completely embed into the two-step iteration of a $\sigma$-closed poset and a $\sigma$-centered poset?

This question allows for slightly more than the vague question of the title; I suppose the first closed poset might be something other than $P(\omega)/\mathrm{fin}$.

One reason to ask this is that a "Yes" would imply that finite products of Miller, Laver, and Mathias forcing are all proper. [Why? Every poset completely embeddable into a closed $\ast$ centered poset is proper, and the class of such posets is closed under finite products.] Note that Spinas has already established the properness of finite powers of Miller & Laver forcing.

Of course, I’d be grateful for whatever references—for negative or positive results—you’re willing to provide.