Serre's problem for multiple conics

We prove the refined Loughran--Smeets conjecture of Loughran--Rome--Sofos for a wide class of varieties arising as products of conic bundles. One interesting feature of our varieties is that the subordinate Brauer group may be arbitrarily large. As an application of our methods, we answer a question of Lenstra by giving an asymptotic for the triples of integers $(a, b, c)$ for which the R\'edei symbol $[a, b, c]$ takes a given value. We also make significant progress on a question of Serre on the zero loci of systems of quaternion algebras defined over $\mathbb{Q}(t_1, \dots, t_n)$.