The conjecture on convergence of the closure-map pushforward measure

Let $\operatorname{Arm}(n,d,w)$ be the space of open $n$-edge arms in $\mathbb{R}^d$ with fixed edgelength $w$, and let $\operatorname{Pol}(n,d,w)$ be the corresponding space of closed polygons. The closure map sends the natural probability measure on $\operatorname{Arm}(n,d,w)$ to a probability measure on $\operatorname{Pol}(n,d,w)$. Closure-map convergence conjecture. As $n\to\infty$, the pushforward measure from $\operatorname{Arm}(n,d,w)$ to $\operatorname{Pol}(n,d,w)$ converges to the standard measure. If true, then for large $n$ random polygons would resemble geometric median closures of random arms, so local phenomena should occur at essentially the same rates in the two spaces.