Iosevich–Wyman polynomial improvement conjecture for products

Let $M:=M_1\times\ldots\times M_n$ be the product of at least two compact Riemannian manifolds without boundary, and let $d$ be its dimension. Denote by $N_M(\lambda)$ the eigenvalue counting function, by $\operatorname{Vol}(M)$ the volume of $M$, and by $\omega_d$ the volume of the $d$-dimensional unit ball. Iosevich–Wyman's conjecture. There exists a $\delta>0$ such that

$$N_M(\lambda)=\operatorname{Vol}(M)\left(\frac{\omega_d}{(2\pi)^d}\right)\lambda^d+O(\lambda^{d-1-\delta}).$$

This conjecture predicts a polynomial improvement over the general $O(\lambda^{d-1})$ remainder in Weyl's law for products of at least two compact manifolds. The cited work proves such an improvement for products of spheres, while the assertion for arbitrary products remains open in the supplied source.