  We draw connections between the various conjectures which are included in G.
R\'emond's generalized Lehmer problems. Specifically, we show that the degree
one form of his conjecture for the multiplicative group is, in a sense, almost
as strong as the strong form. Our insights into the conjectures apply a basic
result on what may be called Diophantine approximation in the metric induced by
the canonical height. We also present a previously unpublished partial result
of R\'emond on his conjectures.
