Semiclassical constant for the eigenvalue sum in three dimensions

For $-\Delta + V$ on $L^2(R^3)$ (units $\hbar^2/(2m) = 1$) with negative eigenvalues $E_j$, prove or disprove $\sum_j \mid E_j \mid \le L^{\mathrm{cl}}_{1,3} \int V_-^{5/2}\, dx$ with $L^{\mathrm{cl}}_{1,3} = 1/(15 \pi^2)$. Dual form: for orthonormal $\phi_1..\phi_N$ in $L^2(R^3)$, $\sum_j \int \mid \mathrm{grad}\, \phi_j \mid^2 \ge (3/5)(6 \pi^2)^{2/3} \int \rho^{5/3}$, $\rho = \sum_j \mid \phi_j \mid^2$. Answer: a proof, or a potential with ratio greater than 1.

A proof of the inequality with constant $1/(15 \pi^{2})$, or an explicit potential exceeding it.