The eventual size bound conjecture for minimal $k$-extendable bipartite graphs

For each $k\geqslant 1$, let $N_k$ be an integer, and let $G$ be a minimal $k$-extendable bipartite graph of order $n$ and size $|E|$. Eventual size bound conjecture. There exists an integer $N_k\leqslant 4k^2+2k$ such that every such graph on $N_k$ or more vertices satisfies

$$|E|\leqslant \frac{(2k+1)(n-2k)}{2}.$$

The authors note that they can construct small counterexamples and therefore impose the lower bound on the order; the conjecture's resolution is not supplied in the text provided.