Sum's conjecture on the recursive construction of minimal generators for polynomial algebras

Let $P_t$ be the polynomial algebra on generators $x_1,\dots,x_t$ over $\mathbb Z_2$, let $\mathcal A_2$ act by Steenrod squares, and let $mathscr C^{\otimes t}_n$ denote the set of $\mathcal A_2$-minimal monomial generators in degree $n$. For $(l,mathscr L)in\roidism \mathcal N_t$, write $r=ell(mathscr L)$, and let $\psi_{(l,mathscr L)}$ and $\widetilde{\Phi_*}$ be the maps defined above. If

$$1 e r=ell(mathscr L)\leq t-1,$$

$x=\prod_{1\leq jleq t-1}x_j^{a_j}in mathscr C^{\otimes(t-1)}_n$, and there exists $u$ with $1\leq uleq r$ satisfying the condition in equation, then

Sum's conjecture.

$$\psi_{(l,mathscr L)}(x)in mathscr C^{\otimes t}_n.$$

Moreover, if $\omega$ is a weight vector of degree $n$, then

$$\widetilde{\Phi_*}\bigl((mathscr C^{\otimes(t-1)}_n)^{\omega}bigr)\subseteq(mathscr C^{\otimes t}_n)^{\omega}.$$

The conjecture proposes that the maps $\psi_{(l,mathscr L)}$ preserve the recursively defined minimal generating sets, including their weight-vector components, and is intended to support the determination of minimal $\mathcal A_2$-generators for $P_t$ in positive degrees. The supplied text gives no resolution, so the conjecture is recorded as open.