The slice-measure inequalities conjecture for two-variable weighted shifts

Let $\mathbf{T}\equiv \mathbf{(}T_{1},T_{2})\in \mathfrak{H}_{0}$ be the $2$-variable weighted shift whose weight diagram is given by Figure 1(i). For each vertical or horizontal slice of $\mathbf{T}$, let its associated subnormal $1$-variable weighted shift have a Berger measure, and let $\psi_k$ denote a linear combination of such Berger measures.

Slice-measure inequalities conjecture. The subnormality of $\mathbf{T}$ is determined by a countable collection of inequalities

$$\psi_k\geq 0.$$

This conjecture proposes a countable measure-theoretic criterion for subnormality using only Berger measures from the one-variable slices of the two-variable weighted shift. The supplied text does not state whether the conjecture has been resolved.