Strong form of Erdős's girth conjecture

A graph $G$ on $n$ vertices with average degree $d$ is almost-regular if every vertex has degree $d+O(1)$.

Strong girth conjecture. For every positive integer $k$, there exists a family of almost-regular graphs $\{G_n\}$ such that $|V(G_n)|=n$, $|E(G_n)|\ge \frac{n^{1+1/k}}{2}$, and $G_n$ is $\{C_4,C_6,\ldots,C_{2k}\}$-free.

This strengthens Erdős's girth conjecture by requiring almost-regularity and forbidding every even cycle through length $2k$. The source states that it is known for $k\in\{2,3,5\}$ via constructions using polarities of generalized polygons; its general validity remains open.