The Steenrod-span conjecture for motivic cohomology operations

Let $F$ be a field. Write $\mathcal{A}^{\star,\star}_{F}$ for the algebra of motivic Steenrod operations and $H\mathbb{F}_{p,F}^{\star,\star}H\mathbb{F}_{p,F}$ for the corresponding bigraded operations on motivic mod-$p$ cohomology. Steenrod-span conjecture. For all fields $F$, the injection

$$\mathcal{A}^{\star,\star}_{F} \subset H\mathbb{F}_{p,F}^{\star,\star}H\mathbb{F}_{p,F}$$

is an equality. This asks whether all motivic mod-$p$ cohomology operations are generated by the motivic Steenrod operations, and is part of the paper's motivation for studying phenomena arising from $p$-adic coefficients.