H-thin implies T-thin conjecture for Khovanov homology

Let $L$ be a link. It is H-thin if its nontrivial Khovanov homology groups lie on two adjacent diagonals. It is T-thin if it is weakly torsion thin and has torsion of order $2$ only.

H-thin implies T-thin conjecture. Every H-thin link is T-thin. In particular, every non-split alternating link is T-thin.

If true, the conjecture would imply that the torsion of every H-thin link is determined by the ranks of its Khovanov homology; the text gives no resolution status.