Polynomial invariants distinguish alternating links within a family

Let $F$ be a family of knots or links, and let $L_1$ and $L_2$ be two alternating, nonisotopic $KL$s belonging to $F$. Let $P$ be a polynomial invariant. Family-distinguishing conjecture. For every such $L_1$, $L_2$, and $P$, one has

$$P(L_1)\neq P(L_2).$$

This conjecture concerns whether polynomial invariants distinguish all nonisotopic alternating knots or links within the same family. The source presents it as a conjecture proposed for families of alternating $KL$s; the supplied material gives no resolution.