Asymptotic separation between systolic bounds for Ramanujan surfaces

Let $g$ denote the genus of a hyperbolic surface, let $\sys(M)$ denote its systole, and let $\Nsmall$ and $\lambda_1(M)$ be the quantities used in the paper. The first quantity below is the smallest upper bound on the systole obtainable from Theorem 3.1, and the second is the smallest lower bound on the systole sufficient to prove $\Nsmall<2$, equivalently $\lambda_1(M)>1/4$, using Theorem 3.2. Asymptotic systolic-gap conjecture. There exist constants $c_1>c_2$ such that these two bounds are respectively of the form

$$2\log(g)+c_j+o(1)\quad\text{as}\quad g\to\infty.$$

The conjecture predicts an asymptotic gap between the systolic upper bound and the lower bound required by the paper's method to establish the Ramanujan property. The source presents this as a conjecture motivated by numerical experiments and asymptotic results; no resolution is given.