Initial degenerations of spinor varieties

We construct closed immersions from initial degenerations of the spinor variety $\mathbb{S}_n$ to inverse limits of strata associated to even $\Delta$-matroids. As an application, we prove that these initial degenerations are smooth and irreducible for $n\leq 5$ and identify the log canonical model of the Chow quotient of $\mathbb{S}_5$ by the action of the diagonal torus of $\operatorname{GL}(5)$.