Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either

1. $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or

2. $f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.

AMS 2000 Mathematics subject classification: Primary 30D35