The domination number of the graph defined by two levels of the $n$-cube, II

Consider all $k$-element subsets and $ell$-element subsets $(k>ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $gamma (G_{k,2})={k+3over 2(k-1)(k+1)}n^2+o(n^2)$.