Does the $\mathsf{TC^0}$ hierarchy collapse? Do we know that the $\mathsf{TC^0}$ hierarchy does not collapse ($\mathsf{TC^0_d} \subsetneq \mathsf{TC^0_{d+1}}$ for all $d$)?

The Zoo entry for $\mathsf{TC^0}$ only mentions the separation between depth 2 and 3.

Also is there a standard reference for the fact that $\mathsf{AC^0_d}$ hierarchy does not collapse?
We know of no good lower bounds (meaning, say, a superpolynomial lower bound for a language in $\mathsf{NEXP}$) for depth 2 threshold circuits (unbounded weights). Depth 3 circuits built from majority gates, i.e. $\mathsf{TC}^0_3$ contains this class, and thus we know no good lower bounds for this class either.