Convexity and endpoint maximum conjecture for spiderweb central configurations

For circles of equal mass, let $n\in\mathbb N$ and $\ell\in\mathbb N_{\geq 2}$, and write

$$a_i=\frac{r_{i+1}(\ell)-r_i(\ell)}{r_1(\ell)},\qquad 1\leq i\leq n-1,$$

for the relative spacing between consecutive circles. Let $a_{i^*}$ be the maximum of the sequence $(a_i)_{1\leq i\leq n-1}$.

Convexity and endpoint maximum conjecture. The sequence $(a_i)_{1\leq i\leq n-1}$ is convex. It is strictly increasing when $\ell=2$, and only when $\ell=2$. Moreover, there exists an increasing function $\mu:\mathbb N\longrightarrow\mathbb N$ such that

$$i^*=\begin{cases}1,&\text{if }\ell\geq\mu(n),\ -1,&\text{if }\ell<\mu(n),\end{cases}$$

and in particular $i^*=1$ whenever $\ell\geq n$.

The convexity forces the maximum to occur at an endpoint, while the claimed function $\mu$ specifies which endpoint occurs. The statement is motivated by numerical experiments; the source does not establish it analytically, so its status remains open.