The power hyponormality conjecture for commuting weighted shifts

Let $(T_{1},T_{2})$ be a commuting two-variable weighted shift, and let $\mathfrak{H}_{\infty}$ denote the class of commuting subnormal pairs.

Power hyponormality conjecture. If

$$(T_{1},T_{2}^{2}),\ (T_{1}^{2},T_{2})\in \mathfrak{H}_{\infty},$$

then

$$(T_{1},T_{2})\in \mathfrak{H}_{\infty}.$$

The preceding theorem shows that, for pairs in the specified class, subnormality of either indicated squared pair is equivalent to subnormality of the original pair; the conjecture asks for the corresponding implication under the stated simultaneous hypothesis.