Temperley–Lieb relations for derived cap and cup functors

For integers $n$ and $k$, let $D^b(\mathcal O_{k,n-k})$ be the bounded derived category of the relevant parabolic category, and define functors

$$\cap_{i,n}:D^b(\mathcal O_{k,n-k})\longrightarrow D^b(\mathcal O_{k-1,n-k-1}),$$ $$\cup_{i,n}:D^b(\mathcal O_{k,n-k})\longrightarrow D^b(\mathcal O_{k+1,n+1-k})$$

by $\cap_{i,n}=\mathcal R\Xi_{n,i}\circ\mathcal R\Gamma_i[1]$ and $\cup_{i,n}=\varepsilon_i\circ\mathcal R\Pi_{n,i}$. Derived Temperley–Lieb relations conjecture. The defining relations of the Temperley–Lieb category, relations (1)–(6) in the source, should hold as natural equivalences for these functors. The first two equivalences follow from results in the paper, while the last relation is identified as $\cap_{i,n+2}\circ\cup_{i,n}\cong Id[1]\oplus Id[-1]$; the remaining relations are not proved there.