Conjecture on the 1-level density beyond finite support

Let $f$ be the test function in the average 1-level density, and let $\sigma$ denote the finite support parameter. Theorem 3/2(3) gives an expansion involving the terms of order $Q^{-1/2}/(\log Q)^k$ for the stated support range. Conjecture on the 1-level density beyond finite support. Theorem 3/2(3) holds for $f$ with arbitrarily large finite support $\sigma$. If true, this would show that the terms of order $Q^{-1/2}/(\log Q)^k$ identified in that theorem persist for arbitrarily large finite support, rather than being artifacts of the restricted support range.