  This paper studies, for a positive integer $m$, the subalgebra of the
cohomology ring of the complex Grassmannians generated by the elements of
degree at most $m$. We build in two ways upon a conjecture for the Hilbert
series of this subalgebra due to Reiner and Tudose. The first reinterprets it
in terms of the operation of $k$-conjugation, suggesting two conjectural bases
for the subalgebras that would imply their conjecture. The second introduces an
analogous conjecture for the cohomology of Lagrangian Grassmannians.
