Global optimizer conjecture for confined elasticae

Consider a rod of length $L$ confined to a ball of radius $R$. Let $r_L=L/(2\pi)$ be the radius of the unconfined elastica, and let $j\in\mathbf{N}$ be a positive integer with $j<r_L/R<j+1$. A confined elastica is a rod configuration satisfying this confinement, and a clew is the corresponding elastica shape denoted by the pair of winding numbers in the claim.

Global optimizer conjecture. There is a number $\xi(j)\in(0,1)$ such that the globally least bent confined elastica is a $j$-$(j+1)$-clew if $r_L/R<j+\xi$, while the global optimizer is the $(j+1)$-$j$-clew otherwise. If $r_L/R<1$, the flat 1-circle is an exactly confined elastica. If $r_L/R=j$, the $j$-fold covered circle is the global elastica.

This conjecture describes the observed transition between two regular families of confined elasticae as the confinement ratio changes. The supplied passage reports numerical observations and does not establish the existence of $\xi(j)$ or the asserted global minimality, so the status remains open.