Symmetry of the Alexander polynomial refinement under orientation reversal

Prove that for every alternating link K, the four-variable Alexander polynomial refinement satisfies P_K(x,y,z,w)=P_K(y,x,w,z).

Background

The polynomial P_K is defined for oriented alternating links and records internal and external activity data of spanning trees of the Tait digraph. Reversing all component orientations exchanges the variables x and y, and z and w, respectively. The paper proves only the partial identities P_K(x,y,1,1)=P_K(y,x,1,1) and P_K(1,1,z,w)=P_K(1,1,w,z), leaving the full four-variable symmetry unresolved for general alternating links.