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 $\mathrm{\Delta }$-matroids. As an application, we prove that these initial degenerations are smooth and irreducible for $n\le 5$ and identify the log canonical model of the Chow quotient of ${\mathbb{S}}_5$ by the action of the diagonal torus of $\mathrm{GL}(5)$.