Does this “Euclidean” sum of reciprocals of primes converge?

Question

The sum $\sum_{k=1}^n1/p_k$, where $p_k$ is the $k$th prime number, is known to diverge as $O(\ln\ln n)$. However, prompted by Euclid, suppose that we restrict our choice of primes to a subset $Q:=\{q_1, q_2,...\}\subset\{p_1,p_2,...\}$, where $q_0:=1$ and $q_{n+1}$ is the least prime divisor of $q_0\cdots q_n+1$ (so that the first few elements of $Q$ are $2,3,7,43,13,53,\dots\,$). The question is: $$\text{does }\,\sum_{q\in Q}\frac1q\; \text{ exist?}$$