Average-degree gap conjecture for simple 3-connected crossing-critical graphs

A graph is $k$-crossing-critical if its crossing number is at least $k$ while deleting any edge reduces the crossing number below $k$. Let $g:\mathbb N\to\mathbb R^+$ be a function. Average-degree gap conjecture. There is a function $g:\mathbb N\to\mathbb R^+$ such that every sufficiently large simple, $3$-connected, $k$-crossing-critical graph has average degree greater than $3+g(k)$. This conjecture predicts a positive gap above average degree $3$ for sufficiently large graphs in each fixed crossing-criticality class; the source presents it as an open structural conjecture.