According to the Complexity Zoo, $\mathsf{Reg} \subseteq \mathsf{NC^1}$ and we know that $\mathsf{Reg}$ cannot count so $\mathsf{TC^0} \not\subseteq \mathsf{Reg}$. However it doesn't say if $\mathsf{Reg} \subseteq \mathsf{TC^0}$ or not. Since we don't know $\mathsf{NC^1}\not\subseteq\mathsf{TC^0}$ we also don't know $\mathsf{Reg} \not\subseteq \mathsf{TC^0}$.
Is there a candidate for a problem in $\mathsf{Reg}$ that is not in $\mathsf{TC^0}$?
Is there a conditional result implying that $\mathsf{Reg} \not\subseteq \mathsf{TC^0}$, e.g. if $\mathsf{NC^1} \not\subseteq \mathsf{TC^0}$ then $\mathsf{Reg} \not\subseteq \mathsf{TC^0}$?