Cubic-cage order conjecture

A cubic graph is a graph in which every vertex has degree $3$. The girth of a graph is denoted by $\operatorname{girth}(G)$, its order by $|V(G)|$, and $c$ denotes a constant.

Cubic-cage conjecture. For infinitely many integers $g$, there exists a cubic graph $G$ such that

$$\operatorname{girth}(G)\geq g\qquad\text{and}\qquad |V(G)|\leq c2^{g/2}.$$

This is proposed as a strengthening of the available general upper bound for the order of cubic cages. It would support the polynomial-expansion path-degeneracy conjecture and further lower-bound conjectures in the paper.