The convergence conjecture for reciprocal prime-gap series

Let a series be formed from reciprocals associated with twin primes and with prime pairs separated by $2k$. The reciprocal-series conjecture. The series of reciprocals of twin primes converges, and the analogous series for prime pairs separated by $2k$ also converge. The source invokes Brun's constant but does not specify the series precisely or provide a resolution status.