Coincidence conjecture for optimal concatenation indices of primitive stepsize schedules

Let $h_{\circ}^{(n)}$ denote the primitive stepsize schedule generated by Definition, let $\Call{ConPP}$ denote the concatenation operation, and let $\mathbf 1$ be the all-ones vector. Coincidence conjecture. For each $n\in\mathbb{N}$, it holds that

$$\mathbf 1^T h_{\circ}^{(n)} = \mathbf 1^T\Call{ConPP}{h_{\circ}^{\lfloor (n-1)/2\rfloor}, h_{\circ}^{\lceil (n-1)/2\rceil}}.$$

Moreover, if $n=p\cdot 2^l-1$, where $p>1$ is an odd integer and $l\in\mathbb{N}$, then

$$\mathbf 1^T h_{\circ}^{(n)} = \mathbf 1^T\Call{ConPP}{h_{\circ}^{((p+i)\cdot 2^{l-1}-1)},h_{\circ}^{((p-i)\cdot 2^{l-1}-1)}},\qquad i\in\{-1,0,1\}.$$

The conjecture formalizes the observed non-uniqueness of optimal concatenation indices: the balanced split is optimal, with additional optimal splits in the stated dyadic cases. Its status is unresolved in the supplied text and is supported only by simple examples and numerical tests.