CH inequality at a given $M$ and $Q$ one could, in principle, choose the highest allowed value of $T$ for each value of $Q$ ($T=T_{\text{max}}(Q)$). We show in Fig. \[S1:Q&lt;M\] the corresponding plot, to be compared with Fig. \[S:Q&lt;M\]. For every $Q<M$ the violation occurs in the same range of angles, namely $\pi/4 \le\Theta\le \pi/2$, because of the following properties of the joint probability distributions: $P^{\Theta}_{1,2}(Q_1,Q_2)=P^{3\Theta}_{1,2}(Q_1,Q_2)=P_{1,-}(Q_1,Q_2)$ for $\Theta=\pi/4$. This implies that ${\cal S}_{CH}(\Theta=\pi/4)=0$, and $P^{\Theta}_{1,2}(Q_1,Q_2)\geq P^{3\Theta}_{1,2}(Q_1,Q_2),P^{\Theta}_{1,-}(Q_1,Q_2),
P^{\Theta}_{-,2}(Q_1,Q_2)$ for $\pi/4 \le\Theta\le \pi/2$. Furthermore, in this specific case of $M=20$, we find that the maximum values of $S$ occurs at $Q=8$.

In Fig. \[Smax:Q&lt;M\] we plot the maximum value of $S$, with respect to $\Theta$ and $T$, as a function of $Q$ for different values of $M$. Several observations are in order. For increasing $M$, the position of the maximum, $Q_{\text{max}}$ is very weakly dependent on $M$. Remarkably, the value of the maximum of the curves does not decreases exponentially, but rather as $1/M^2$. Despite the exponential suppression of the joint probability with $M$, the extent of the maximal violation scales with $M$ much slowly (polynomially).

It may be useful to look at the same situation from a different perspective. Given a 