  The k-monotone classes of densities defined on (0, \infty) have been known in
the mathematical literature but were for the first time considered from a
statistical point of view by Balabdaoui and Wellner (2007, 2010). In these
works, the authors generalized the results established for monotone (k=1) and
convex (k=2) densities by giving a characterization of the Maximum Likelihood
and Least Square estimators (MLE and LSE) and deriving minimax bounds for rates
of convergence. For k strictly larger than 2, the pointwise asymptotic behavior
of the MLE and LSE studied by Balabdaoui and Wellner (2007) would show that the
MLE and LSE attain the minimax lower bounds in a local pointwise sense.
However, the theory assumes that a certain conjecture about the approximation
error of a Hermite spline holds true. The main goal of the present note is to
show why such a conjecture cannot be true. We also suggest how to bypass the
conjecture and rebuild the key proofs in the limit theory of the estimators.
