the convex sum of a single PR-box plus up to eight more local deterministic strategies. For instance, any correlation violating the CHSH inequality can be written as $$\label{defbox}
        p(ab|xy)=c_0 PR(ab|xy) + \sum_{\alpha,\beta,\gamma=0}^{1} c_{\alpha\beta\gamma} L_{\alpha\beta\gamma}(ab|xy),$$ where $$\label{defPR}
    PR(ab|xy)=PR^{000}(ab,xy),$$ $$\label{deflocal}
    L_{\alpha\beta\gamma}(ab|xy)=L_{\alpha\beta\gamma(\alpha \gamma\oplus \beta)}(ab|xy),$$ and $c_0 + \sum_{\alpha,\beta,\gamma=0}^{1} c_{\alpha\beta\gamma}=1$, $0\leq c_0\leq 1$ and $0\leq c_{\alpha\beta\gamma}\leq 1$ $\forall$ $\alpha$, $\beta$, and $\gamma$. In this case $\mathrm{CHSH}(PR)=4$ (maximal violation) and $\mathrm{CHSH}(L_{\alpha\beta\gamma})=2$ (these local points saturate the local bound of the inequality). Furthermore, any other nonlocal distribution can be achieved via local reversible transformations (relabelings) over such distribution. In this sense, in the CHSH scenario it is thus sufficient to consider only and we will do so in what follows. To simplify the notation, from now on we will refer to the local deterministic strategies as $L_i$ (where might assume eight different values $i=1,\dots,8$). Here, we follow the notation in [@PhysRevA.99.032106], as shown in Table \[Li\_Labg\].

$\begin{array}{|c|c|c|c|c|c|c|c|c|}
 \hline
 L_i & L_1 & L_2  & L_3 &L_4 & L_5 & L_6 & L_7& L_8  \\
\hline
 L_{\alpha\beta\gamma} & L_{101} & L_{111} & L_{001} & L_{011} & L_{110} & L_{100} & L_{000} & L_{010} \\
 \hline
\end{array}$

As will be described in more details below, we are interested 