Monotonicity conjecture for optimal Variant II CPC compositions

Let $J>1$. For a Variant II concentric permutation code, let $\boldsymbol{\boldsymbol{ u}}=(n_1, obreak\thinspace\bot)$ be its composition, where $n_i$ denotes the multiplicity associated with index $i$, and let $\boldsymbol{\boldsymbol{ u}}$ be optimized over admissible compositions. Assume that $E[\eta_\ell]$ is convex in $\ell$, namely

$$E\left[\eta_{\ell+2}\right]-2E\left[\eta_{\ell+1}\right]+E\left[\eta_{\ell}\right]\geq 0,\qquad 1\leq \ell\leq n-2.$$

Monotonicity conjecture. If $J>1$ and $E[\eta_\ell]$ is convex in $\ell$, then the optimal multiplicities $n_i$ for Variant II CPCs increase monotonically with $i$.

This conjecture is an analogue of a necessary condition known for optimal compositions of ordinary permutation codes. The paper notes that the convexity condition holds for a large class of source distributions, including Gaussian sources, and that the conjecture would substantially reduce the search space for optimal compositions; only a restricted-codeword version is proved.