Uniform boundedness conjecture for bilinear quadratic functionals over disjoint strips

Let ${\mathcal S}$ be a collection of pairwise disjoint strips, and let $\Pi_{\mathcal S}$ denote the associated bilinear quadratic functional. For exponents $p_1,p_2\in[2,\infty)$, define $p$ by

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$$

Uniform boundedness conjecture. For every $\epsilon>0$, uniformly over all collections ${\mathcal S}$ of disjoint strips,

$$\|\Pi_{\mathcal S}\|_{L^{p_1}\times L^{p_2}\to L^p}\lesssim (\sharp\mathcal S)^\epsilon.$$

The authors suggest that the factor $(\sharp\mathcal S)^\epsilon$ might even be removable. This is presented as an expected optimal range by analogy with the linear theory, but no resolution is given in the supplied text.