The strip-density classification conjecture

Let $\mathbb{R}^2$ have the strip density, equal to $1$ in the strip $\{|y|\leq 1\}$ and equal to $\lambda>1$ outside it. The relevant candidate sets are classified as types (i), (ii), and (iii). Strip-density classification conjecture. There exists a value $v_0>\pi$ such that the isoperimetric sets are balls of type (i) for areas less than $\pi$, sets of type (ii) for areas in $[\pi,v_0]$, and sets of type (iii) for areas greater than $v_0$. The source says that the theorem proves everything except the elimination of type (iv), so the stated classification is not established there; no later resolution is supplied.