The $k$-linear oscillatory integral estimate for Hrmnder-type operators

Let $1\leq k\leq n$, and let $(T_1,\dots,T_k)$ be a $\nu$-transverse $k$-tuple of H"ormander-type operators of the same signature $\sigma$. Thus, if $G_j$ denotes the generalised Gauss map associated to $T_j$, then

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

for all points in the relevant supports. The $k$-linear conjecture. For every $\lambda\geq1$, the estimate

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

holds whenever $p\geq\bar p(n,\sigma,k)$. The conjecture is the stronger multilinear statement for which the paper's $k$-broad theorem serves as a substitute: some extreme cases follow from existing linear, multilinear, or bilinear estimates, while the remaining cases are presented as new and the asserted exponent range is sharp.