The local-knot frequency conjecture for random arms and polygons

In dimension $3$, let $K^\text{Arm}(n,w,k,r)$ be the probability that a length-$k$ arc of a random open arm is an $r$-local knot, and let $K^\text{Pol}(n,w,k,r)$ be the corresponding probability for a random closed polygon. Here an $r$-local knot is a subsegment that intersects the boundary of a radius-$r$ ball only at its endpoints and forms a knotted ball-arc pair with that ball. Local-knot frequency conjecture. For small $r$, large $n$, and $k\ll n$,

$$K^\text{Arm}(n,w,k,r)\simeq K^\text{Pol}(n,w,k,r).$$

This predicts that local knotting occurs at essentially the same rate in open arms and closed polygons in the stated asymptotic regime, motivated by the conjectured convergence of the closure-map pushforward measure.