The conjectured coinvariant dimensions in the degree $2^{s+3}+2^s-2$ family

Let $k=\mathbb Z_2$, let $P_4=k[x_1,x_2,x_3,x_4]$, and let $P((P_4)_n^*)$ be the primitive part of the dual degree-$n$ component under the Steenrod algebra action. For a positive integer $s$, consider the $GL_4(k)$-coinvariant space

$$k\otimes_{GL_4(k)}P((P_4)_{2^{s+3}+2^s-2}^*).$$

Coinvariant-dimension conjecture. Its dimension is

$$\dim k\otimes_{GL_4(k)}P((P_4)_{2^{s+3}+2^s-2}^*)= \begin{cases} 0&\text{if }s=1,\\ 1&\text{if }s=2,\\ 2&\text{if }s=3,\\ 1&\text{if }s=4,\\ 2&\text{if }s\geq5. \end{cases}$$

This proposed dimension formula follows from the Kameko-map decomposition together with the preceding kernel-invariant conjecture and known transfer information. It remains open in the source.