Friedman's adjacency and Markov operator conjecture for random graph coverings

Let $\Omega$ be a finite connected graph, let $\Gamma$ be a random $n$-covering of $\Omega$, and let $\lambda_A(\Gamma)$ and $\lambda_M(\Gamma)$ denote the largest absolute values of the non-trivial eigenvalues of the adjacency operator and Markov operator, respectively. Let $\rho_A(\Omega)$ and $\rho_M(\Omega)$ be the spectral radii of the corresponding operators on the universal cover of $\Omega$. Friedman's graph-covering spectral conjecture. For every $\varepsilon>0$, asymptotically almost surely,

$$\lambda_A(\Gamma)<\rho_A(\Omega)+\varepsilon$$

and likewise

$$\lambda_M(\Gamma)<\rho_M(\Omega)+\varepsilon.$$

The adjacency-operator assertion is attributed to Friedman, and the Markov-operator assertion is proposed as an extension. The source presents the combined statement as a conjecture; the paper establishes weaker nearly optimal bounds rather than these exact inequalities.