Discrete total variation under the multinomial, Part 2: variance under the uniform null

For the counts $(X_1,\dots,X_n)\sim\mathrm{Mult}(N;1/n,\dots,1/n)$ of $N$ observations in $n$ ordered bins we study the variance of the discrete total variation $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$. An exact decomposition splits $\mathrm{Var}(d)$ into per-edge, adjacent-pair and disjoint-pair terms and