Let $n \geq 20$ and $\kappa \in \mathbb{N}$ with $3\kappa \leq n + 2$. Then every gate of a reduced matrix-symmetric circuit of orbit size $< \binom{n}{\kappa}$ has a support of size $\leq \kappa - 1$, i.e. the circuit has support size $\leq \kappa - 1$.

In the paper, the orbit size is bounded by a function $f \colon \mathbb{N} \to \mathbb{N}$, and $\kappa_f(n) = \min\{r \in \mathbb{N} : \binom{n}{r} > f(n)\}$. The statement there is that, if $n \geq 20$ and $3\kappa_f(n) \leq n + 2$, every gate of a reduced matrix-symmetric circuit of orbit size $\leq f(n)$ has a minimal support of size $\leq \kappa_f(n) - 1$. This is the case $\kappa = \kappa_f(n)$ of the statement above. Conversely, the statement above for any $\kappa$ follows from the paper's for the constant function $f$ with value the orbit size, since then $\kappa_f(n) \leq \kappa$. As $2(\kappa - 1) < n$, the minimal supports exist and are contained in every support of size $\leq \kappa - 1$, so bounding the support size is the same as bounding the size of the minimal supports.