Higher-dimensional logarithmic-phase fractal uncertainty conjecture

For $n=2$, let $X,Y\subset B_{\mathbb R^2}(0,1)$ be $\nu$-porous on scales $h$ to $1$. Let $U\subset\mathbb R^{2n}$ be open, let $b\in C_c^\infty(U)$, and define

$$\mathcal B_hf(x)=(2\pi h)^{-n/2}\int_{\mathbb R^n}e^{i\Phi(x,y)/h}b(x,y)f(y)\,dy,$$

where

$$\Phi(x,y)=\log|x-y|.$$

Higher-dimensional logarithmic-phase FUP conjecture. For each $\nu>0$ there exists $\beta=\beta(\nu)>0$ such that

$$\|\mathbf 1_X\mathcal B_h\mathbf 1_Y\|_{L^2(\mathbb R^n)\to L^2(\mathbb R^n)}=\mathcal O(h^\beta)\quad\text{as }h\to0.$$

The claim concerns a genuinely higher-dimensional phase for which the standard reduction to the Fourier transform may lose information; it is proposed as a generalization relevant to higher-dimensional hyperbolic manifolds.