Classical zero-curvature trilinear Hilbert transform conjecture

Let $\gamma(t)=(a_1t,a_2t,a_3t)$, where $a_1,a_2,a_3$ are nonzero and pairwise distinct, so that $\gamma$ is a nondegenerate line in $\mathbb{R}^3$. For $\vec f=(f_1,f_2,f_3)$ and $\vec p=(p_1,p_2,p_3)$, define $\\|\vec f\\|_{L^{\vec p}}:=\prod_{j=1}^3\\|f_j\\|_{L^{p_j}}$. Classical zero-curvature trilinear Hilbert transform conjecture. The operator $\mathbf T_\gamma$ obeys

$$\\|\mathbf T_\gamma(\vec f)\\|_{L^{p_4}}\lesssim_{\vec p,\gamma}\\|\vec f\\|_{L^{\vec p}}$$

for every $\vec p\in(1,\infty]^3$ satisfying $\sum_{j=1}^3 1/p_j=1/p_4$ and $1/2<p_4<\infty$. This is presented as a major open problem in Fourier analysis: the boundedness of the classical, zero-curvature trilinear Hilbert transform remains unresolved in the stated range.