This is related to an older question.

Let $(X,\tau)$ be a topological space. Trivially, the indiscrete topology $\{\emptyset, X\}$ is a connected subtopology of $\tau$.

Is there a connected topology $\tau_c\subseteq \tau$ on $X$ such that whenever $\tau_c'$ is a connected topology on $X$ such that $\tau_c\subseteq\tau_c'\subseteq \tau$ then $\tau_c=\tau_c'$?