  Laptev and Safronov conjectured that any non-positive eigenvalue of a
Schr\"odinger operator $-\Delta+V$ in $L^2(\mathbb R^\nu)$ with complex
potential has absolute value at most a constant times
$\|V\|_{\gamma+\nu/2}^{(\gamma+\nu/2)/\gamma}$ for $0<\gamma\leq\nu/2$ in
dimension $\nu\geq 2$. We prove this conjecture for radial potentials if
$0<\gamma<\nu/2$ and we `almost disprove' it for general potentials if
$1/2<\gamma<\nu/2$. In addition, we prove various bounds that hold, in
particular, for positive eigenvalues.
