The non-resonant boundedness conjecture for the curved trilinear Hilbert transform and curved n-linear maximal operator

Let $H_{n,\vec{\alpha},\vec{\beta}}$ and $\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}$ denote, respectively, the curved trilinear Hilbert transform and the curved $n$-linear maximal operator in the non-resonant regime. For exponents $p_1,\ldots,p_n$ and $r$ satisfying

$$\sum_{j=1}^n \frac{1}{p_j}=\frac{1}{r},\qquad 1<p_j<\infty,\qquad \frac{1}{n}<r<\infty,$$

The non-resonant boundedness conjecture. Up to end-points, both operators have the boundedness range

$$H_{n,\vec{\alpha},\vec{\beta}},\,\mathcal{M}_{n,\vec{\alpha},\vec{\beta}}:L^{p_1}(\mathbb{R})\times\cdots\times L^{p_n}(\mathbb{R})\longmapsto L^r(\mathbb{R}).$$

This is the maximal boundedness range asserted for these operators in the non-resonant regime, extending earlier results and addressing the quasi-Banach range $r<1$. The supplied text gives no resolution status beyond the assertion, so the claim is recorded as open.