Weyl law improvement conjecture for products of spheres

Let $M=M_1\times\cdots\times M_k$ be a product of compact Riemannian manifolds, where $M_i$ has dimension $d_i>0$, and let $\Delta$ be the Laplace–Beltrami operator for the product metric. Write $|d|=\dim M=d_1+\cdots+d_k$, and suppose that the eigenvalues of $\Delta$, repeated with multiplicity, are $-\lambda_j^2$. Define

$$N(\lambda):=\#\{j:\lambda_j\leq\lambda\}.$$

Weyl law improvement conjecture. There exists $\delta>0$ such that

$$N(\lambda)=\frac{|B_{|d|}|}{(2\pi)^{|d|}}\operatorname{vol}(M)\lambda^{|d|}+O\left(\lambda^{|d|-1-\delta}\right).$$

The conjecture asserts a power improvement over the general Weyl remainder for products of compact Riemannian manifolds, motivated by the improved bounds known for the flat torus and by the structure of products of manifolds. The supplied context does not establish the conjecture, and its resolution status is therefore left open.