The conjectured coinvariant dimensions in the degree $2^{s+4}+2^{s+1}-3$ 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

$$k\otimes_{GL_4(k)}P((P_4)_{2^{s+4}+2^{s+1}-3}^*).$$

Coinvariant-dimension conjecture. This space is trivial if $s=2$ and has dimension $1$ if $s\neq 2$.

This is one of the remaining cases in the investigation of Singer's conjecture for rank $4$; the stated dimensions are proposed on the basis of the preceding computations, and their general validity remains open.