Exponential average-availability conjecture for local girth in planar graphs

Let $G$ be a planar graph and let $f$ be a local girth function for $G$. For an operation sequence $\sigma$, define its average availability by

$$\overline a(\sigma)=\prod_{i=1}^n a(\sigma_i)^{1/n},$$

where the availability of $\mathsf{Del}(G,f,v)$ is $f(v)+1$ and the availability of $\mathsf{DelSave}(G,f,v,w)$ is $f(v)-f(w)$. A sequence is legal if each operation is legal, and it removes every vertex when it certifies weak $f$-degeneracy.

Exponential average-availability conjecture. There exists a constant $c>1$ such that, for every planar graph $G$ and local girth function $f$ for $G$, there is a legal sequence of operations $\sigma$ with average availability at least $c$ that removes every vertex of $G$.

Together with the proposition preceding the conjecture, this would yield exponentially many local-girth correspondence colourings of planar graphs: the number of colourings would be at least $\overline a(\sigma)^{v(G)}$, hence at least $c^{v(G)}$. The supplied text gives no evidence that the conjecture has been resolved.