Ore's periodicity conjecture for the minimum size of k-critical graphs

Let $f_k(n)$ denote the minimum number of edges in a $k$-critical graph on $n$ vertices, where a graph is $k$-critical if its chromatic number is $k$ and every proper subgraph is $(k-1)$-colorable.

Ore's conjecture. If $k\geq4$, then

$$f_k(n+k-1)=f_k(n)+(k-1)\left(\frac{k}{2}-\frac{1}{k-1}\right).$$

This conjecture predicts the exact periodic behavior of the minimum edge count of $k$-critical graphs. The paper's abstract says that Kostochka and Yancey proved an asymptotic form of Ore's conjecture, while the exact assertion is not identified there as resolved.