Extension of Bourgain's Return Times theorem below $p=2$

Let ${\bf X}=(X,\Sigma,\mu,\tau)$ be a dynamical system, let $1<p,q\leq\infty$, and suppose that

$$\frac{1}{p}+\frac{1}{q}<\frac{3}{2}.$$

Return Times conjecture. For every function $g\in L^q(X)$ there is a universal set $X_0\subseteq X$ with $\mu(X_0)=1$ such that, for every second dynamical system ${\bf Y}=(Y,\mathcal F,\nu,\sigma)$, every $f\in L^p(Y)$, and every $x\in X_0$, the averages

$$\frac{1}{N}\sum_{n=0}^{N}g(\tau^n x)f(\sigma^n y)$$

converge for $\nu$-almost every $y$.