Greene's branched-double-cover conjecture for alternating links

Let $L_1$ and $L_2$ be links whose branched double covers are homeomorphic.

Greene's conjecture. If a pair of links have homeomorphic branched double-covers, then either both are alternating or both are non-alternating.

This conjecture concerns whether alternation is determined by the homeomorphism type of a link's branched double cover. The source attributes the conjecture to Greene and gives no resolution, so its status remains open.