Independent mutual-visibility coloring conjecture for subdivided complete graphs

Let $K_n$ be the complete graph on $n$ vertices, let $S(K_n)$ denote its subdivision graph, and let $\chi_{\mu_i}(G)$ be the independent mutual-visibility chromatic number of a graph $G$. Let $\rho(n)$ be the parameter defined in the paper.

Independent mutual-visibility coloring conjecture. For every positive integer $n$,

$$\chi_{\mu_i}\big(S(K_n)\big)=\rho(n)+1.$$

The preceding results establish that the ordinary mutual-visibility chromatic number of $S(K_n)$ is either $\rho(n)$ or $\rho(n)+1$, while the conjectured exact value for the independent version remains open.