  We show that the decay of approximation numbers of compact composition
operators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, which
was left open in the cited work. We also prove the optimality of a result of
O.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on
$\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in
$\mathcal{D}$.
