Let us continue this discussion in chat.

My plot shows that it is not a convex function of $x$ even when all $p_i = x + 0.1$ and $q_i = x$, $n = 3$.

Are you sure that the objective is convex?

Once we prove that the objective is convex, the rest follows easily from symmetry. The optimal solution must be invariant under the permutation group (acting on the indices of $p_i$ and $q_i$) and "reflections": $(p, q) \mapsto (1 - q, 1- p)$. The only fixed point is your solution.

(1) Your note is correct; (2) $d$ only determines the distribution of the first coordinates of points $x_i$; (3) it's not necessarily the case that the property holds for all $n$ starting with some $n_0$, since both the rejection probability and $p(n)$ depend on $n$

Corrected the issue with strict/non-strict inequalities.

In 3d, the only polytope like this is the triangular prism.

Accordingly, there might be a “counterexample” $x$ to RH in the model in $\cal U$, which is a non-standard natural number.

“$\cal U$ contains an isomorphism between the natural numbers of $\cal U$ and the natural numbers in a model of ZFC inside $\cal U$” — this is not necessarily the case; even if $\cal U$ is a transitive model of ZFC, the submodel inside $\cal U$ might have non-standard natural numbers.