For closed $k$-Schur Katalan functions $\fgλ{k}$ with $k$ a positive integer and $λ$ a $k$-bounded partition, Blasiak, Morse and Seelinger proposed the alternating dual Pieri rule conjecture and the $k$-branching conjecture. In the present paper, we positively prove the first one for large enough $k$ and for strictly decreasing partitions $λ$ respectively, as well as the second one for strictly decreasing partitions $λ$.