Discrete total variation under the multinomial, Part 3: expectation under a general distribution

For multinomial counts $X\sim\operatorname{Mult}(N;\mathbf p)$ on $n$ ordered bins and a reference law $\mathbf r$ we study the expectation of the reference-centered discrete total variation $d_{\mathbf r}(X)=\sum_{j=1}^{n-1}|(X_{j+1}-Nr_{j+1})-(X_j-Nr_j)|$, the natural extension of the comb-test st