 Solution

False. Take $X=\mathbb{R}$ and $A=B=\{0\}$. Both complements are disconnected: their negative and positive halves form a separation. Butis path-connected, hence connected. To see path connectedness directly, move any nonzero point radially to its normalization on the unit circle; join the two normalized points by a circular arc; then reverse the radial path for the second point. Every segment of this path avoids the origin. Thus disconnected coordinate complements do not force a disconnected complement of their product.