Extend vanishing above the Chow line to MGL and the motivic sphere at the characteristic

Determine whether the vanishing result above the Chow line—namely, kgl^{i,j}(X) = 0 for i > 2j for all smooth S-schemes X—extends to algebraic cobordism cohomology MGL^{i,j}(X) and, further, to the motivic sphere spectrum 1_S (in the modified form of Haine–Pstrągowski, Lemma 3.3.4), at the characteristic. Establish these vanishing properties without relying on the Hopkins–Morel–Hoyois equivalence, which is not currently known to hold at the characteristic.

Background

The authors’ construction of weight filtrations leverages a key connectivity/vanishing input for effective K-theory (kgl), namely that kgl^{i,j}(X) vanishes for i > 2j. This vanishing enables splitting and boundedness properties necessary for the weight structure.

They explicitly state uncertainty about whether analogous vanishing results extend to MGL-motives or the motivic sphere at the characteristic. Away from the characteristic, such results rely on the Hopkins–Morel–Hoyois equivalence, but this equivalence is not known at the characteristic, leaving the extension as an open problem.