A full-support optimal product setting law exists

Let $S$ be a Bell experiment with exactly two parties, exactly two settings per party, and exactly two outcomes at every setting. Thus its setting tuples and measured outcome tuples are pairs in $\{0,1\}^2$, and a potential assignment $v$ specifies a bit $v_i(b)$ for each party $i\in\{0,1\}$ and setting $b\in\{0,1\}$; let $\mathcal V$ be the set of these assignments. Let $Q_a(x)$ be a real table such that each $Q_a$ is a probability distribution, its marginals on every subset of parties agree whenever the settings on that subset agree, and it has a finite dimensional quantum representation: there exist positive local dimensions, a positive semidefinite joint matrix $\rho$ of trace $1$, and positive semidefinite local measurement matrices summing to the identity for each party and setting, with $Q_a(x)$ equal to the real part of the trace of their tensor product times $\rho$. Write $\Delta(E)$ for the probability distributions on a finite set $E$, and define $P_p(a,x)=\sum_{v:\,v_i(a_i)=x_i\ \forall i}p(v)$ for $p\in\Delta(\mathcal V)$. For two finite probability distributions $q,r$, let $D(q\Vert r)=\infty$ if some coordinate has $q>0$ and $r=0$; otherwise let it be $\max\{0,\sum_{z:q(z)\ne0}q(z)\log_2(q(z)/r(z))\}$, regarded as an extended nonnegative real. Define

Then there exists $s\in\Delta(\{0,1\}^2)$ such that $s(a)>0$ for every setting tuple $a$, such that $s(a_0,a_1)=u_0(a_0)u_1(a_1)$ for some $u_0,u_1\in\Delta(\{0,1\})$, and such that

The supremum includes product distributions with zero coordinates, although the asserted attaining distribution has every coordinate strictly positive.