The weak 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 $\psi^*(\iota_p)\in \mathrm{H}^p(X)$.

Weak-form conjecture. As graded algebras, one has

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

This conjecture proposes that, after quotienting by nilpotent elements, the cohomology of the homotopy fiber splits as the tensor product of the corresponding quotient of the base cohomology and the cohomology of $K_{p-1}$. The preceding results provide a monomorphism from the first factor and an epimorphism onto the second factor, but whether they combine into this graded-algebra description is left open.