On continuous isomorphisms from $\sigma$-compact paratopological groups onto topological groups

In this short note we give a negative answer to the following open question: \emph{Let $X$ be a $\sigma$-compact paratopological group. Does there exist a continuous isomorphism of $X$ onto a topological group $G$?} Specifically, we construct a family of $\sigma$-compact paratopological groups for which a continuous isomorphism onto a topological group $G$ exists if and only if $G$ carries the trivial topology.