The first collapsing-function identity for worms and spiders

Let $\uppsi_0$ be the ordinal collapsing function and let $\cond_{\top}$ denote the corresponding worm/spider collapsing operation. The first collapsing-function identity. One has

$$\uppsi_0{\Upomega^\omega 1}=\cond_{\top}\left({0 \atopwithdelims \langle \rangle{\Upomega^\omega 1}}\top\right).$$

This identity is presented as a conjecture because the paper concludes with it as a proposed translation between the two notation systems; the supplied context gives no proof or resolution.