What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?

Of course, there must also be such a morphism in every larger dimension - just take $X \times \mathbb P^n$.

As described in this excellent question, $d\geq2$. As described in the accepted answer, $d\leq 6$. We can improve that to $d \leq 5$ by noting that the E7 lattice also produces a nonsingular hypersurface, because the unique potential singular point over $\mathbb F_2$ fails to lie on the hypersurface.

But that still leaves a lot of uncertainty! Can anyone clarify?

Here is an auxiliary question, which I think might prove easier to answer:

What is the smallest dimension of an $X$ satisfying those conditions that is also the flag variety of a reductive group?