Modified Newton distance conjecture for oscillatory integral operators

Let $S(x,z)\in\mathfrak S^m\mathbb R^{n_X+n_Z}$, and let $\delta_{mod}(S)$ denote its modified Newton distance, defined by

$$\delta_{mod}(S)=\sup\left\{\delta\left(S\left(Ax,Bz\right)\right): A\in GL(n_X),\,B\in GL(n_Z)\right\}.$$

Modified Newton distance conjecture. If $S\in\mathfrak S^m\mathbb R^{n_X+n_Z}$, then

$$\lVert T_\lambda\rVert\le C\lambda^{-1/(2\delta_{mod}(S))}\left(\log(\lambda)\right)^p$$

for some $p\ge 0$. The conjecture proposes a general decay bound governed by the largest Newton distance obtained after independent linear changes of variables in $x$ and $z$, extending the relationship known in lower-dimensional and previously treated cases; its general validity is not established in the supplied context.