distance between consecutive eigenvalues for the laplacian on cubes

Question

The asymptotic expansion of the eigenvalues of the Dirichlet Laplacian on a cube $[0,\pi]^d$ is given by Weyl's asymptotic, namely it starts with $$ \lambda_n = C(d) n^{2/d}+o(n^{2/d}). $$ This fact is very general, and true for most well behaved domains (with a factor being the volume of the domain). In this particular case, where it is a matter sorting $$ (k_1^2+\cdots+k_d^2)_{k_1\geq0,\cdots,k_d\geq0} $$ in increasing order, since the eigensolutions are explicitly known ($\sin(k_1x_1)\cdots\sin(k_dx_d)$) is more known? More precisely, if we call $(\lambda_{n_i})$ the subsequence of distinct increasing eigenvalues, is it known that $$\lambda_{n_{i+1}}-\lambda_{n_{i}}> C n_i^{\frac2d-1}$$ or another lower bound, this (perhaps naive) one is what would come out if the Weyl asymptotic was exact.