Strict growth conjecture for prime knot counts by crossing number

Let $n$ be a positive integer. Let $P$ be a prime knot projection and $K$ a prime knot. Let $c(P)$ be the number of double points of $P$ and $c(K)$ the minimum number of crossings of $K$. Define

$$\mathcal{K}_n=\{K\mid c(K)=n\},\qquad \mathcal{P}_n=\{P\mid c(P)=n\}.$$

For a set $S$, let $|S|$ denote its cardinality.

Strict growth conjecture for prime knots. If $3<n<m$, then

$$|\mathcal{K}_n|<|\mathcal{K}_m|.$$

This is identified in the source as a famous conjecture concerning the number of prime knots with a given crossing number. The same candidate span also records analogous assertions for prime knot projections and the inequality $|\mathcal{K}_n|\leq|\mathcal{P}_n|$, but the first assertion is the explicitly attributed conjecture; its resolution is not indicated in the supplied text.