The linear lower-bound conjecture for small eigenvalues of cusped hyperbolic surfaces

Let $\mathcal{M}_{g,n}$ denote the moduli space of hyperbolic surfaces of signature $(g,n)$, and call an eigenvalue of the Laplacian small when it is less than $\frac{1}{4}$. Here $g$ is the genus and $n$ is the number of cusps.

Linear small-eigenvalue conjecture. There are constants $a,b>0$ such that, whenever

$$n>a\left(g+1\right),$$

every surface in $\mathcal{M}_{g,n}$ has at least

$$b\left(2g+n-2\right)$$

small eigenvalues.

The conjecture would remove the logarithmic loss in the paper's unconditional lower bound and establish an almost linear number of small eigenvalues whenever the number of cusps dominates the genus. The paper presents it as unresolved; its status is therefore open.