Small eigenvalues of random 3-manifolds

We show that for every $g\ge 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)\mathrm{/}\mathrm{v}\mathrm{o}\mathrm{l}(M{)}^2$.