Positive boundedness conjecture for case iv singular Brascamp–Lieb data

Let $\mathbf{H}_\mathbf{M}$ be a singular Brascamp–Lieb datum, where $\mathbf{M}$ is as in case iv) of Theorem Hclass. Let $\mathbf{p}=(p_1,p_2,p_3)$ satisfy $1<p_i<\infty$ for each $i$ and

$$\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=1.$$

Positive boundedness conjecture. Every such datum $\mathbf{H}_\mathbf{M}$ is $\mathbf{p}$-bounded.

The conjecture would, by the theorem cited in the surrounding discussion, imply that the conditions in Theorem Hclass are sufficient for the relevant positive boundedness results. The supplied text gives no resolution, so the conjecture remains open.