The two-thirds regularity conjecture for prime knots

Let $\operatorname{cr}(X)$ denote the crossing number of a knot $X$. A prime knot $P$ is $\lambda$-regular if

$$\operatorname{cr}(K)\geq \lambda\cdot\operatorname{cr}(P)$$

whenever $P$ is a factor of a knot $K$. Two-thirds regularity conjecture. Each prime knot is $\frac23$-regular. The paper records Lackenby's result that every knot is $\frac1{152}$-regular and studies the consequences of the stronger conjecture; no resolution is supplied.