We propose a $q$-analog of classical plethystic conjectures due to Foulkes.
In our conjectures, a divided difference of plethysms of Hall-Littlewood
polynomials $H_n(\mathbf{x};q)$ replaces the analogous difference of plethysms
of complete homogeneous symmetric functions $h_n(\mathbf{x})$ in Foulkes
conjecture. At $q=0$, we get back the original statement of Foulkes, and we
show that our version holds at $q=1$. We discuss further supporting evidence,
as well as various generalizations.Comment: 13 page