 Solution

A connected component $U$ of ${\rho }_A(x)$ meets the relatively clopen subset ${\rho }_B(x)$ either everywhere or nowhere: its intersection and its complement would otherwise disconnect $U$. If it meets nowhere, $U\subseteq {\sigma }_B(x)$. If it meets everywhere, $U$ is a connected subset of ${\rho }_B(x)$, and any larger connected subset of ${\rho }_B(x)$ would be a connected subset of ${\rho }_A(x)$ larger than its component $U$. Thus $U$ is either a whole component of ${\rho }_B(x)$ or lies wholly in ${\sigma }_B(x)$.