Extended exponential tail bound for 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

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

Extended tail-bound conjecture. As $x\to\infty$,

$$\pi_k(x;h)\ll x\exp\left(-\frac{k}{\lambda e}\right).$$

This extends the paper's conditional tail bound from $k\ll(\log h)^{1-\delta}$ to the larger range $k\ll(\log h)^2$. The source presents it as a prediction and gives no resolution status.