$ACC^0$ implementation of a boolean function

Consider the symmetric boolean function $$F(x_1,\dots,x_n)=1\iff\sum_{i=1}^nx_i\mbox{ is a square}.$$

It is implementable in $$TC^0$$.

Is there an $$ACC^0$$ implementation?

The reason I ask is there seems to be few tricks which can be deployed.

1. Square is sum of consecutive odd numbers.

2. Square is $$0$$ or $$1$$ mod $$4$$ (checkable by $$\oplus$$ gates).

3. Squares are the set of the integers having odd number of divisors.

Few are available in https://proofwiki.org/wiki/Category:Square_Numbers.

But is there a finite number of tricks which capture the function and place it in $$ACC^0$$?

Suppose $$x \in \{0,\dots,2n\}$$. Then we have $$((2n)^2 - n) + x$$ is square if and only if $$x=n$$. This is easily seen to imply that the square function is complete for $$\mathrm{TC}^0$$ under $$\mathrm{AC}^0$$ Turing reductions.