Norine–Thomas linear-density conjecture for degree-three vertices in minimal bricks

A brick is a 3-connected bicritical graph. A brick $G$ is minimal if $G-e$ is not a brick for every edge $e$ of $G$. Write $V(G)$ for its vertex set and let $|V(G)|$ denote the number of vertices.

Norine–Thomas conjecture. There exists a constant $\alpha>0$ such that every minimal brick $G$ has at least $\alpha|V(G)|$ vertices of degree three.

Norine and Thomas established that every minimal brick has at least three vertices of degree three and conjectured a positive linear proportion. The statement remains open according to the supplied context.