The first negative eigenvalue conjecture for Siegel Hecke eigenforms

Let $F$ be an arbitrary Siegel Hecke eigenform of degree $2$ and weight $k\geq 20$, with Hecke eigenvalues $\lambda_F(n)$ and analytic conductor $Q_F$. For $\epsilon>0$, write $n$ for the least positive integer such that $\lambda_F(n)<0$.

First negative eigenvalue conjecture. For every $\epsilon>0$, one has

$$n\ll_{\epsilon} Q_F^{1/2+\epsilon},$$

where the implied constant is absolute apart from its dependence on $\epsilon$.

The paper proves a stronger bound for Yoshida lifts under a condition on the conductors of the underlying forms. This conjecture proposes the analogous convexity-scale bound for arbitrary degree-two Siegel Hecke eigenforms, and the source suggests that an exponent strictly smaller than $1/2$ might also be possible.