Jutila-type gap conjecture for zeros of half-integral weight cusp-form $L$-functions

Let $N\in\mathbb{N}$ and let $f(z)=\sum_{n=1}^{\infty}a_f(n)e^{2\pi inz}\in S_{k+\frac{1}{2}}\left(\Gamma_0(4N)\right)$ satisfy $f|W_{4N}=f$ or $f|W_{4N}=-f$, and suppose that the coefficients $a_f(n)$ are either real or purely imaginary.

Jutila-type gap conjecture. For every $\epsilon>0$, there exists $T_0(\epsilon)$ such that, for all $T\geq T_0(\epsilon)$, $L(s,f)$ has a zero of the form

$$s=\frac{k}{2}+\frac{1}{4}+i\tau$$

with

$$\tau\in[T,T+T^{\frac{1}{3}+\epsilon}].$$

This conjecture proposes the half-integral-weight analogue of a gap theorem of Jutila for zeros of $L$-functions attached to integral-weight cusp forms. Its validity is motivated by arguments using Rankin-type mean-value estimates and would require suitable Voronoi-type transformation formulas; the source does not report a proof or disproof.