Adapted-function reduction for quadratic-manifold restriction estimates

Let $fbf{Q}(fxi)=(Q_1(fxi),\ldots,Q_n(fxi))$ be an $n$-tuple of real quadratic forms in $d$ variables, and let

$$\mathcal{M}=\{(\xi,\mathbf{Q}(\xi)):\xi\in[0,1]^d\}\subset\mathbb{R}^{d+n}.$$

For a rectangle $B\subset[0,1]^d$, call $f$ adapted to $B$ if $f(\xi)=\psi(L\xi)$, where $\psi$ is a fixed non-negative smooth cutoff equal to $1$ on $[\frac14,\frac34]^d$ and supported in $[\frac18,\frac78]^d$, and $L$ satisfies $[0,1]^d=L(B)$. Adapted-function reduction conjecture. The following are equivalent: the $L^q\to L^p$ restriction estimate holds for all measurable functions $f$, and it holds for functions adapted to any rectangles. This would reduce the restriction problem for quadratic manifolds to testing estimates on geometrically adapted functions; the source provides no resolution of the equivalence.