 Solution

Intersect a connected component $U$ of ${\rho }_A(x)$ with the clopen subset ${\rho }_B(x)$. Connectedness makes the intersection either empty or all of $U$. In the empty case $U\subseteq {\sigma }_B(x)$. In the other case $U\subseteq {\rho }_B(x)$; a connected set in ${\rho }_B(x)$ containing $U$ is also connected in ${\rho }_A(x)$, so maximality prevents it from being larger. Thus $U$ is a component of ${\rho }_B(x)$, or $U\subseteq {\sigma }_B(x)$.