Grassmannian permutations avoiding identity

A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid the identity permutation. In particular, we count the Grassmannian permutations of size $m$ avoiding the identity of size $k$, thus solving a conjecture made by Weiner. We also study avoidance of identity in special classes such as odd Grassmannian permutations and Grassmannian involutions.