Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges?

Can one compute a few digits based on euristic considerations or plausible conjectures about distributions of primes and prime gaps? I think it may be a bit less that 0.63, but I'm not at all confident.