Cherednik's Riemann-hypothesis conjecture for the Hilbert L-function

Let $(C,0)$ be a planar curve singularity, and let $L(q,t,0)$ be the specialization at $a=0$ of its Hilbert $L$-function. A $t$-zero is a complex number $\xi$ such that $L(q,\xi,0)=0$. Cherednik's Riemann-hypothesis conjecture. Any $t$-zero $\xi$ of $L(q,t,0)$ satisfies

$$|\xi|=\frac{1}{\sqrt{q}},\qquad\forall\,0<q\leq\frac{1}{2}.$$

This is an analogue of a Riemann hypothesis for the zeros of the specialized Hilbert $L$-function. The source identifies it as the main version of I. Cherednik's conjecture but supplies no evidence of a resolution.