Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?

Let $$R$$ be a spectrum. Assume that $$R$$ is bounded-below. Then we can “even-ify” $$R$$: cone off some generating set of the first nonvanishing odd-dimensional homotopy group of $$R$$. Do this repeatedly. In the end you have a spectrum $$R^{ev}$$ with even-dimensional homotopy. Moreover, you only added even cells, so if $$R$$ had even cells to begin with, then $$R^{ev}$$ has even cells.

If $$R$$ is a ring spectrum, then so is $$R^{ev}$$, by obstruction theory.

Question 1: If $$R$$ is a connective ring spectrum with even cells, then is the homotopy of $$R^{ev}$$ zero-divisor-free?

Question 2: If $$R$$ is a ring spectrum with even homotopy and even cells, then is $$\pi_\ast R$$ zero-divisor-free?

Notes:

• I’m actually not quite sure how to ensure that $$R_\ast$$ is a commutative ring, so perhaps the title question — asking if $$R_\ast$$ is in fact polynomial — doesn’t quite make sense.

• For example, when $$R$$ is the sphere, I think we get $$R^{ev} = MU$$. At any rate, if we work $$p$$-locally and take $$R$$ to be the $$p$$-local sphere, then I’m certain I’ve been told that $$R^{ev} = BP$$. So an abstract proof that even-ification produces polynomial homotopy groups might provide some alternate way to think about the Milnor/Quillen computation of the homotopy groups of $$MU$$.

• If we do this unstably, then the possible output spaces were completely classified by Steve Wilson in his PhD thesis. I think the theorem says they are all products of spaces of the form $$\Omega^{\infty-n}BP$$. At any rate, the list is very finite.

• Note that if we don’t require $$R$$ to have even cells, then there are Eilenberg-MacLane counterexamples (Just take $$HR$$ where $$R$$ is an even-graded ring with zero-divisors). But I’m pretty sure that an Eilenberg-MacLane spectrum never has even cells.

Question 3: Is there a classification of (ring) spectra with even homotopy and even cells analogous to Wilson’s classification of spaces with even homotopy and even cells?

Let $$G$$ be any discrete group, and let $$MU[G] = MU \otimes \Sigma^\infty_+ G$$ be the associated group algebra over $$MU$$. Additively, $$MU[G] \simeq \bigoplus_{g \in G} MU$$, and so it has both even homotopy and is built from even cells - it is (one choice of) its own evenification. However, its homotopy typically has zero-divisors (if $$g^n = 1$$, then $$(1-g)(1+g+\dots+g^{n-1}) = 0$$).
Another example is given by the trivial square-zero extension $$MU \oplus \Sigma^{2n} MU$$, whose coefficient ring is $$MU_*[x] / x^2$$ for a generator $$x$$ in degree 2n.
(The identification of the evenification of $$BP$$ is due to Priddy, "A cellular construction of BP and other irreducible spectra". In the integral case there is not usually a canonical minimal construction of $$R^{ev}$$ like there is in the $$p$$-local case.)