Monotonicity conjecture for optimal Variant I CPC compositions

Let $J>1$, and define

$$\mathcal{S}_1=\{1,2,\ldots,\lfloor K/2\rfloor\},\qquad \mathcal{S}_2=\{\lfloor K/2\rfloor+1,\lfloor K/2\rfloor+2,\ldots,K\}.$$

For Variant I concentric permutation codes, let $n_i$ denote the multiplicity associated with index $i$. Assume that $E[\xi_\ell]$ is convex on $\mathcal{S}_1$ and concave on $\mathcal{S}_2$.

Variant I monotonicity conjecture. If $J>1$, then the optimal $n_i$ for Variant I CPCs increase monotonically with $i\in\mathcal{S}_1$ and decrease monotonically with $i\in\mathcal{S}_2$.

This is presented as a straightforward extension of the preceding Variant II conjecture. It is intended to reduce the search space for optimal compositions; the source does not provide a proof or establish that the required restriction on codewords preserves optimality.