Kuperberg's uniform Poisson conjecture for primes in short intervals

Let $\pi(t)$ denote the number of primes at most $t$. For a real parameter $\lambda=(\log x)^{o(1)}$ and an integer $k\ll (\log_2 x)^2$, count integers $n\leqslant x$ according to the number of primes in the interval $(n,n+\lambda\log x]$. Kuperberg's conjecture.

$$\big|\{n\leqslant x:\pi(n+\lambda\log x)-\pi(n)=k\}\big|\sim x\frac{e^{-\lambda}\lambda^k}{k!}\quad(x\rightarrow\infty).$$

This is a uniform Poisson law for prime counts in short intervals, extending the fixed-parameter heuristic to slowly growing $\lambda$ and moderately growing $k$; the source gives no resolution.