Bourgain–Sarnak–Rudnick-type conjecture for shrinking almost-round regions

Assume $n=3$ and $m\in S$. Let $A_m$ be a family of sets on the sphere depending on $m$. Suppose there are absolute constants $c_1$ and $c_2$ such that $A_m$ contains a spherical cap of radius $c_1\sigma(A_m)^{1/2}$ and is contained in a spherical cap of radius $c_2\sigma(A_m)^{1/2}$. Suppose also that, for some $\delta>0$,

$$r(m)^{-1+\delta}\lesssim \sigma(A_m)\lesssim r(m)^{-\delta}.$$

Bourgain–Sarnak–Rudnick-type conjecture. For every $\epsilon>0$,

$$\int_{SO(3)}\Delta^2(gA_m)\\,dg=O\left(m^\epsilon r(m)^{-1}\sigma(A_m)\right).$$

This extends the Bourgain–Sarnak–Rudnick estimate for shrinking spherical caps and segments to more general shrinking regions, and is intended to improve the arithmetic coupling result in dimension three. Its resolution status is not specified in the supplied text.