The conjectured kernel-invariant dimensions for Kameko's squaring map

Let $k=\mathbb Z_2$, let $P_4=k[x_1,x_2,x_3,x_4]$, and write $Q^{\otimes 4}=k\otimes_A P_4$. For a positive integer $s$, Kameko's squaring operation is the $kGL_4(k)$-module epimorphism

$$[\overline{Sq}^{0}]_{2^{s+3}+2^s-2}:Q^{\otimes4}_{2^{s+3}+2^s-2}\longrightarrow Q^{\otimes4}_{2^{s+2}+2^{s-1}-3}.$$

Let $\operatorname{Ker}[\overline{Sq}^{0}]_{2^{s+3}+2^s-2}$ denote its kernel. Kernel-invariant dimension conjecture. The invariant space

$$\left(\operatorname{Ker}[\overline{Sq}^{0}]_{2^{s+3}+2^s-2}\right)^{GL_4}$$

is trivial if $s=1,2$ and has dimension $1$ if $s\geq3$.

This conjecture concerns one of the remaining $t=3$ cases in the rank-$4$ analysis and is intended to determine the corresponding invariant and coinvariant spaces. Its status is unresolved in the source.