Short intervals with a given number of primes

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is asymptotically equal to $\lambda^me^{-\lambda}/m!$ as $x$ tends to infinity. We show that the number of such $n$ is at least $x^{1 - o(1)}$.