Dvořák–Feghali linear diameter conjecture for list recolouring of planar graphs

Let $G=(V,E)$ be a planar graph on $n$ vertices. A list assignment $L$ assigns a set $L(v)$ of colours to each vertex $v\in V$, and let $\mathcal{G}(G,L)$ be the graph whose vertices are the proper $L$-colourings of $G$, with edges between colourings differing on exactly one vertex. Dvořák–Feghali's conjecture. If $|L(v)|\geq 10$ for every $v\in V$, then

$$\mathcal{G}(G,L)\text{ has diameter }O(n).$$

A linear bound is known for ordinary recolouring of planar graphs with ten colours, but the corresponding list-colouring statement remains the subject of the cited conjecture; the paper addresses it with further results.