Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an increasing function of $k$.

Unless I've made a mistake, the conjecture holds if you replace the binomial with its normal approximation and $p < n/(m+n)$. This suggests the conjecture should be true for large $m$ and $n$. But is it true for all $m$ and $n$, i.e. for small values? How large can $p$ be?