L^p estimates for semi-degenerate simplex multipliers

A multi-index $\vec{\alpha} \in \mathbb{R}^n$ is called semi-degenerate when it satisfies the semi-degeneracy conditions defined in the paper, and $C^{\vec{\alpha}}$ denotes the corresponding simplex multiplier class. Semi-degenerate simplex multiplier conjecture. For every semi-degenerate $\vec{\alpha} \in \mathbb{R}^n$, $C^{\vec{\alpha}}$ satisfies some $L^p$ estimates. This conjecture asks whether the mapping properties established for particular semi-degenerate simplex multipliers extend to every semi-degenerate multi-index; the paper gives motivation for the question but does not establish the asserted estimates in full generality.