Twin-prime occurrence in every interval between consecutive prime squares

Let $p_n$ denote the $n$th prime. Twin-square-interval conjecture. For every integer $n\geq 3$, there exists at least one pair of twin primes lying in the interval

$$(p_{n-1}^2,p_n^2).$$

This is a stronger interval-by-interval assertion than the infinitude of twin primes and is supported in the paper by a probabilistic analysis and a computation through $n=5\cdot 10^3$.