Froese's resonance counting conjecture

Let $V$ be a super-exponentially decreasing potential, so that its Fourier–Laplace transform $\widehat V$ is entire, and let $F$ be its Froese function,

$$F(z)=\widehat V(2z)\widehat V(-2z)+1.$$

Assume that $\widehat V$ has completely regular growth. Let $A(R,\theta,\varphi)=\{re^{i\alpha}\in\mathbf{C}:r\leq R,\ \alpha\in[\theta,\varphi]\}$, let $n(R,\theta,\varphi)$ count the resonances in this sector, and let $N(R,\theta,\varphi)$ count the zeros of $F$ there. Froese's conjecture. In the lower-half plane $\mathbf{C}_-$,

$$\left|n(R,\theta,\varphi)-N(R,\theta,\varphi)\right|=o(R^\rho).$$

Here $\rho$ is the order of $\widehat V$ (of normal type). Froese proved the conjecture for a large class of potentials, including Gaussians, but the statement is not established in the full generality above.