Poisson conjecture for the distribution of primes in short intervals

Let $x>1$, let $h=\lambda\log x$, where $\lambda=o((\log x)^\varepsilon)$ for every $\varepsilon>0$, and let $k\ll(\log h)^2$. Define

$$\pi_k(x;h):=\#\{n\leq x:\pi(n+h)-\pi(n)=k\}.$$

Poisson conjecture. As $x\to\infty$,

$$\pi_k(x;h)\sim x\frac{\lambda^k e^{-\lambda}}{k!}.$$

This predicts that the Poisson model remains valid for substantially growing values of $k$, including the range $k\ll(\log h)^2$. The source says that existing upper and lower bounds are far from matching this prediction.