Badakhshian–Katona–Tuza asymptotic conjecture for the domination number of $G_{k,2}$

Let $G_{k,2}$ be the bipartite graph with vertex classes $\binom{[n]}{k}$ and $\binom{[n]}{2}$, where a $k$-element set $S$ is adjacent to a 2-element set $T$ exactly when $T\subset S$, and let $\gamma(G)$ denote the domination number of a graph $G$. Badakhshian–Katona–Tuza conjecture. For every fixed integer $k\geq 3$,

$$\gamma(G_{k,2}) = \frac{k+3}{2(k-1)(k+1)}n^{2} + o(n^{2}).$$

Badakhshian, Katona and Tuza had previously established upper and lower bounds for this domination number; the conjecture predicts its precise leading asymptotic term.