Inversion monotonicity for 1324-avoiding permutations in a near-quadratic range

Let A(n,k) denote the number of 1324-avoiding permutations of length n with k inversions. We prove A(n,k) ≤ A(n+1,k) for 0 ≤ k ≤ floor(n2 / (20 log2(n+1))). We also prove the next fixed line beyond the previously known range: A(n,2n+7) ≤ A(n+1,2n+7) for every n ≥ 1. Together with Meng's theorem for k ≤ 2n+6, this gives inversion monotonicity throughout the larger of these two ranges. The complete Claesson–Jelínek–Steingrímsson conjecture remains open. The near-quadratic proof combines the Linusson–Verkama injection with an encoding of the remaining permutations and explicit estimates for integer partitions. The fixed-line proof uses increasing inflations of reduced permutations and exact finite enumeration. The accompanying archive contains the manuscript and LaTeX source, exact certificates, frozen source code, enumeration output, verification scripts, attribution, and a SHA-256 manifest.