We know that languages defined by (poly-sized) Boolean formulae equals $\mathbf{NC}^1$: that Boolean formulae can be simulated in $\mathbf{NC}^1$ was shown by Brent/Spira [B,S], and the converse is quite trivial (simply duplicate the circuit whenever there is a ''collision''). A proof can be found, e.g., in this lecture note by Pitassi (Theorem 3).
Boolean formulae (since the underlying graph is a binary tree) can be thought of as circuits with treewidth $1$, where treewidth of a circuit can be defined as the treewidth of the graph obtained by ignoring the direction of the wires. Gal and Jang [GJ] extended the result in [B,S] by showing that a circuit of size $s$ with treewidth $w$ can be simulated in depth $w\log(s)$. My question is whether the converse is also true: e.g., whether $\mathbf{NC}^2$ can be simulated using circuits with treewidth $\log(s)$ with only a polynomial blow-up (and more generally, whether $\mathbf{NC}^i$ can be simulated using circuits with treewidth $O(\log^{i-1}(s))$).
Intuitively, this should hold (by following the tree decomposition and then using the duplicating trick). But I could not find any resource regarding this (which makes me wonder whether it is false and there is a blow-up in some step). Any pointers would be appreciated!
[B]: Brent, The Parallel Evaluation of General Arithmetic Expressions. Journal of the ACM, 1974.
[GJ]: Gàl and Jang, A Generalization of Spira’s Theorem and Circuits with Small Segregators or Separators, Inf. Comput., 2016.
[S]: Spira, On time-hardware complexity tradeoffs for Boolean functions. Hawaii Symp. on System Sciences, 1971.
