Failure of the square-root pattern for the three-legged spider constant

Let $C_n$ denote the spider constant for a spider with $n$ legs, so that the explicitly computed values are $C_0=1$, $C_1=\sqrt{2}$, and $C_2=\sqrt{3}$. Failure of the square-root pattern. The pattern $C_n=\sqrt{n+1}$ does not hold for $n=3$. The claim concerns the first case beyond the explicitly solved values and is based on simulation approaches; the source does not provide a proof or a replacement value for $C_3$.