The strong form of the mod-2 cohomology conjecture for decomposable Postnikov towers

Let $E$ be the homotopy fiber of a map $\psi:X\rightarrow K_p$, where $K_p=K(\mathbb F_2,p)$, and write $\mathrm{H}^*(X)$ for mod-$2$ singular cohomology. Let $\iota_p\in \mathrm{H}^p(K_p)$ be the fundamental class, and let $\sqrt{A}$ denote the quotient of a graded algebra $A$ by its ideal of nilpotent elements. The map is assumed to represent a decomposable element $\psi^*(\iota_p)\in \mathrm{H}^p(X)$.

Strong-form conjecture. As unstable algebras, one has

$$\sqrt{\mathrm{H}^*(E)}\cong \sqrt{\mathrm{H}^*(X)/(\psi^*(\iota_p))}\otimes \mathrm{H}^*(K_{p-1}).$$

This strengthens the weak form by requiring the proposed isomorphism to preserve the unstable-algebra structure, not merely the graded-algebra structure. The preceding monomorphism and epimorphism motivate the conjecture, while the existence of such an unstable-algebra decomposition remains open.