Skein relations for tangle Floer homology

In a previous paper, V'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{C}\mathrm{T}}(T)$. If $L$ is obtained by gluing together $T_1,\dots ,T_m$, then the knot Floer homology $\widehat{\mathrm{H}\mathrm{F}\mathrm{K}}(L)$ of $L$ can be recovered from $\widetilde{$\mathrm{C}\mathrm{T}$} (T_1), \dotsc, \widetilde{$\mathrm{C}\mathrm{T}$} (T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology.