Two members $U_{v}$ and $V_{w}$ of this covering intersect if and only if

$$v\in(\operatorname{carrier}_{\Delta^{\vee}}w)^{\vee}\iff w\in(\operatorname{carrier}_{\Delta}v)^{\vee}\iff\langle v,w\rangle=1,$$

in which case they intersect in the contractible set $\operatorname{star}_{\Sigma}((v,w))$. So the claimed homotopy equivalence follows from the nerve lemma (cf. e.g. [Bjö95, Thm. 10.6]). ∎