One local law matches any three setting pairs

Let $S$ be a Bell experiment with exactly two parties and exactly two settings per party, with a positive finite number of outcomes at every setting. A setting tuple is $a=(a_0,a_1)\in\{0,1\}^2$, an outcome tuple at $a$ belongs to the product $X_a$ of the two corresponding outcome sets, and a potential assignment $v\in\mathcal V$ chooses an outcome $v_i(b)$ for each party $i$ and each of its settings $b$. Let $Q_a(x)$ be a real table whose rows are nonnegative and sum to $1$, whose marginal probabilities on any subset of parties agree whenever the settings on that subset agree, and for which there exist positive local dimensions, a positive semidefinite joint matrix $\rho$ of trace $1$, and positive semidefinite local measurement matrices $M_{i,b,u}$ summing to the identity over $u$, with $Q_a(x)=\operatorname{Re}\operatorname{tr}((M_{0,a_0,x_0}\otimes M_{1,a_1,x_1})\rho)$. For every specified setting tuple $o\in\{0,1\}^2$, there exists a function $p:\mathcal V\to\mathbb R$ with $p(v)\ge0$ for every $v$ and $\sum_v p(v)=1$, such that for every setting tuple $a\ne o$ and every $x\in X_a$,

There is no condition on this equality at the specified tuple $o$; agreement there is also permitted.