Temperley–Lieb relations for parabolic cap and cup functors

Let $\mathcal O^{k,n-k}$ be the relevant parabolic category, let $\zeta_n:\mathcal O^{k,n-k}_1\stackrel{\cong}{\longrightarrow}\mathcal O^{k-1,n-k-1}$ be the Enright–Shelton equivalence, and define cap and cup functors by

$$\cap_{i,n}=\zeta_n\circ\tau_2^1\circ\tau_3^2\circ\dots\circ\tau_{i-1}^{i-2}\circ\tau_i^{i-1}\circ T^i,$$ $$\cup_{i,n}=T_i\circ\tau_{i-1}^i\circ\tau_{i-2}^{i-1}\circ\dots\circ\tau_2^3\circ\tau_1^2\circ\zeta_{n+2}^{-1}.$$

Parabolic Temperley–Lieb relations conjecture. These functors should satisfy the natural isomorphisms corresponding to the defining relations of the Temperley–Lieb category, with $q=-1$ in the last relation. The first two relations are immediate from Enright–Shelton and the paper's results, and the last follows from translation-functor identities; proving the remaining four requires a deeper understanding of $\zeta_n$.