Exercise concerning quaternion algebras

Here’s a fun problem that came up in a talk by Jacob Tsimerman on Monday concerning some joint work with Andrew Snowden:

Problem: Let \(D/\mathbf{Q}(t)\) be a quaternion algebra such that the specialization \(D_t\) splits for almost all \(t\). Then show that \(D\) itself is split.

As a comparison, if you replace \(\mathbf{Q}\) by \(\overline{\mathbf{Q}}\), then although the condition that \(D_t\) splits becomes empty, the conclusion is still true, by Tsen’s theorem.

This definitely *feels* like the type of question which should have a slick solution; can you find one?