Dickson-image conjecture for the third Hilbert-series term when $q=n=2$

Let $S=\mathbb{F}_q[x_1,x_2]$, let $Q=S/(x_1^{q^m},x_2^{q^m})$, and let $G=Gl_2(\mathbb{F}_q)$. In the case $q=n=2$, consider the third term, referred to in the source as the $k=3$' term, of the conjectured Hilbert series of $Q^G$. An invariant of $Q$ is an **image of a Dickson invariant polynomial** if it is the image under $S\to Q$ of an element of $S^G$. **Dickson-image conjecture.** For arbitrary $m$, the $k=3$' term corresponds to invariant polynomials of $Q$ that are images of Dickson invariant polynomials in $S$. The conjecture is intended to complement the preceding $m=n+1$ case and would support the claim that, over $\mathbb{F}_2$ with $n=2$, all invariants arise as images of Dickson invariants; the source does not establish it.