Barrington–Straubing–Thérien conjecture on CC-circuit lower bounds

Let $m \in \mathbb{N}$, and let $(C_n)_{n \in \mathbb{N}}$ be a family of bounded-depth $\operatorname{CC}[m]$-circuits computing $\operatorname{AND}$.

Barrington–Straubing–Thérien conjecture. There exists $q>0$ such that $C_n$ has size

$$2^{\Omega(n^q)}.$$

The conjecture asserts a strong lower bound for computing conjunction using only modular gates. The surrounding discussion identifies this as an open question; for non-prime $m$, the source gives substantially smaller upper bounds, so the precise optimal exponent remains an additional issue.