The $k$-linear H"ormander conjecture

Let $1\leq k\leq n$, and let $\mathbf{T}^{\lambda}=(T_1^{\lambda},\dots,T_k^{\lambda})$ be a $\nu$-transverse $k$-tuple of H"ormander-type operators with positive-definite phase functions. Here $\nu$-transverse means that, for the associated generalized Gauss maps $G_j$, one has

$$\left|\bigwedge_{j=1}^k G_j(x;\omega_j)\right|\geq\nu$$

for all $(x;\omega_j)$ in the supports of the amplitudes, and define $\bar p(k,n):=2(n+k)/(n+k-2)$. $k$-linear H"ormander conjecture. For every $p\geq\bar p(k,n)$ and $\varepsilon>0$,

$$\left\|\prod_{j=1}^k|T_j^{\lambda}f_j|^{1/k}\right\|_{L^p(\mathbb{R}^n)}\lesssim_{\varepsilon,\nu,\phi}\lambda^{\varepsilon}\prod_{j=1}^k\|f_j\|_{L^2(B^{n-1})}^{1/k}$$

holds for all $\lambda\geq1$. This is presented as a natural generalization of Bennett's conjecture for Fourier extension operators; the source does not state a resolution, so the conjecture remains open.