Polynomial fertility conjecture for reduced knot shadows

Let $S$ be a reduced shadow with $n$ crossings. Say that $S$ resolves into a knot type if some assignment of overcrossings and undercrossings to its crossings produces a diagram of that knot. There are positive constants $c$ and $\alpha$ such that every reduced shadow with $n$ crossings resolves into at least

$$cn^\alpha$$

distinct knot types.

This conjecture asks for a universal polynomial lower bound on the number of knot types represented by any reduced shadow. It strengthens the preceding linear-growth question and remains open in the supplied text.