Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations

Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\le a(n+1,k)$ for $k\le2n-7$. We study the obstruction beyond that line: the residuals $\mathcal R_{δ,n}$, namely the indecomposable, non-almost-decomposable avoiders at d