An unoriented skein exact triangle in unoriented link Floer homology

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to $I^{\sharp}$ of the knot in our 2-surgery exact triangle.