Conditional on a strong form of the Hardy-Littlewood conjectures, the Poisson tail conjecture for primes in short intervals holds for slowly growing λ with a phase transition identified for larger λ.
Kuperberg,Sums of singular series with large sets and the tail of the distribution of primes, Q
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The Poisson Tail Conjecture for Primes in Short Intervals
Conditional on a strong form of the Hardy-Littlewood conjectures, the Poisson tail conjecture for primes in short intervals holds for slowly growing λ with a phase transition identified for larger λ.