Explicit $H_3$ relation in the diagrammatic Hecke category

We compute the missing three-color relation in type $H_3$ as an explicit $R$-linear combination of double leaves. The expression has $43424$ nonzero polynomial coefficients with scalars in $\mathbb{Z}[\phi ]$, where $\phi =(1+\sqrt{5})\mathrm{/}2$. We verify the relation using exact arithmetic in the computer algebra system GAP. This completes the diagrammatic presentation of the Hecke category of Elias and Williamson.