Pointwise convergence of double Birkhoff averages for \b7R^m-systems

Let $(X,\mathcal{X},\mu,(T_g)_{g\in\mathbb{R}^{m}})$ be an $\mathbb{R}^{m}$-system, meaning a measure-preserving $\mathbb{R}^{m}$-action. For $f_1,f_2\in L^{\infty}(\mu)$ and $\alpha_1,\alpha_2\in\mathbb{R}^{m}$, consider the double Birkhoff average

$$\lim_{T\to\infty}\frac{1}{T}\int_0^T f_1(T_{\alpha_1t}x)f_2(T_{\alpha_2t}x)\,dt.$$

Double Birkhoff averages conjecture. Every $\mathbb{R}^{m}$-system is good for double Birkhoff averages: for every $f_1,f_2\in L^{\infty}(\mu)$ and every $\alpha_1,\alpha_2\in\mathbb{R}^{m}$, the displayed limit exists for $\mu$-almost every $x\in X$. If true, this would make the preceding characterization a complete answer to the weak equidistribution problem for dilated measures. The conjecture is stated in the source as an open question in ergodic theory.