Laptev–Safronov conjecture for complex Schrödinger operators

Let $d\geq2$ and $q\in(d/2,d]$. Let $V$ be a bounded complex-valued potential on $\mathbb{R}^d$, and let $z$ be an eigenvalue of $-\Delta-V$ with $z\in\mathbb{C}\setminus[0,\infty)$. Here $A\lesssim_{d,q}B$ means that $A\leq C_{d,q}B$ for a constant depending only on $d$ and $q$. Laptev–Safronov conjecture. Every such eigenvalue satisfies

$$|z|^{q-\frac{d}{2}}\lesssim_{d,q}\int_{\mathbb{R}^d}|V|^q.$$

This conjecture seeks a complex-potential analogue of the Keller–Lieb–Thirring bound that remains uniform for eigenvalues approaching the positive real axis. The statement is attributed to Laptev and Safronov; the supplied source does not indicate whether it has been resolved.