Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Østvær generalized those results to a base field of characteristic $p$, as long as the coefficients are mod \$\ell\$ with \$\ell\neq p\$. The case \$\ell=p\$ remains conjectural.

In joint work with Markus Spitzweck, we show that over a base field of characteristic $p$, the conjectured form of the mod \$p\$ dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins--Morel--Hoyois isomorphism, a statement about the algebraic cobordism spectrum \$MGL\$.