Proof: In fact, the Sturmfels–Tevelev multiplicity formula is the special case where $X=U^{\prime}$ is a closed subvariety of $T^{\prime}$ (see [Gu12], Theorem 13,17, for a proof in our setting deducing it from the original sources). In the general case, we conclude that

$$({\rm Trop}(\psi))_{*}({\rm Trop}(\varphi_{*}^{\prime}(U^{\prime})))={\rm Trop}(\psi_{*}((\varphi^{\prime})_{*}(U^{\prime})))={\rm Trop}(\varphi_{*}(U^{\prime})).$$

Since $U^{\prime}$ is dense in $U$, the claim follows. $\square$