Upper $k$-star-forming sets and $k$-independence in bipartite graphs

Sahbi conjectured that the $k$-independence number $\beta_k$ and the upper $k$-star-forming number \SF_k agree on bipartite graphs, and in particular on chain graphs, and asked whether \beta_3=\SF_3 on triangle-free graphs. For every $k\ge3$ we give a connected bipartite graph on $3k+3$ vertices with $\beta_k=2k+2$ and \SF_k\ge3k. On chain graphs, however, \beta_k=\SF_k for every $k$. We also show, answering another of his questions, that deciding whether a bipartite graph has a minimal $2$-star-forming set with at least $t$ vertices is NP-complete.