Nilpotence filtration conjecture for the mod-2 cohomology of highly connected spaces

Let $X$ be a $2d$-connected space, with $d\geq 1$, and let $M=\widetilde H^*(X;\mathbf F_2)$ be its reduced mod-$2$ cohomology. Assume that all cup products in $M$ are trivial, that $M\in\mathcal{N}il_d$, that $M/M_{2d}\in\mathcal U_1$, and that, for every integer $s\leq d$, the unstable module $\bar T^s(M)$ is finite-dimensional in each degree. Nilpotence filtration conjecture. Then $M\in\mathcal{N}il_{d+1}$. This is presented as a cautious generalization of the preceding theorem and is described as seemingly within reach; no resolution is supplied here.