Steenrod-generation conjecture for the invariants $a_{m,2,k'}$

Fix $m$ and let

$$A=\{a_{m,2,k'}:0\leq k'\leq\frac{q^m-q}{q-1}\},$$

and

$$B=\{a_{m,2,k'}:k'=0\text{ or }k'=1+\frac{q^t-1}{q-1}\text{ for }1\leq t\leq m-1\}.$$

Here the $a_{m,2,k'}$ are the invariants introduced for the $n=2$ case, and Steenrod operations act on them. Steenrod-generation conjecture. Every element of $A$ can be generated by Steenrod operations from elements of $B$: $a_{m,2,0}$ generates itself and $a_{m,2,1}$; if $1\leq t<m-1$ and $k'=1+\frac{q^t-1}{q-1}$, then $a_{m,2,k'}$ generates $a_{m,2,l}$ for $k'\leq l\leq k'+q^t$; and for $t=m-1$ it generates $a_{m,2,l}$ for $k'\leq l\leq k'+q^t-1$. This conjecture would reduce the invariant family to a smaller set of Steenrod generators, but the source gives no resolution.