Parallel crossing number conjecture for knots and links

Let $L$ be any knot or link. Let $\operatorname{PC}(L)$ denote its parallel overcrossing number and $\operatorname{Cr}(L)$ its crossing number.

Parallel crossing number conjecture.

$$\operatorname{PC}(L)=\operatorname{Cr}(L).$$

This is presented as a weaker consequence of the conjecture that the asymptotic crossing number equals the crossing number. The paper does not report a proof, and the equality would improve its ropelength estimates.