Universality of lower-order density terms for composite levels

Let $N$ be a level whose prime factors tend to infinity, and consider the $n$-level density of the associated holomorphic cusp newforms. The lower-order terms are compared with those for prime level $N$.

Lower-order universality conjecture. As long as the prime factors of $N$ tend to infinity sufficiently fast relative to the reciprocal of the error scale being considered, the lower-order terms of the $n$-level density agree with those in the prime-level case. New factors can appear when one prime factor tends to infinity more slowly than that reciprocal scale, for example at rates $\log(N)$, $\log^2(N)$, or $\log^3(N)$.

The paper proves this phenomenon to the stated precision in several prime and semiprime scenarios, including agreement when both factors grow sufficiently quickly, and observes different lower-order terms when one factor is fixed. The conjecture concerns the corresponding behavior for general levels and density levels.