On Chow and Picard groups of Richardson varieties

We show that if $\mathrm{\Pi }\subseteq G\mathrm{/}P$ is a projected Richardson variety, then the degree map $\mathrm{P}\mathrm{i}\mathrm{c}(\mathrm{\Pi })\to A_1(\mathrm{\Pi }{)}^{\vee }$ is injective. Moreover, it is an isomorphism in characteristic zero and in all but finitely many positive characteristics. We conjecture that it is an isomorphism in every characteristic. Our proofs are geometric, using a linear paving, topological properties of klt singularities, and reduction to characteristic zero.