Infinite admissible lengths conjecture for consecutive prime sums

Let $p_n$ denote a prime number, and for each integer $k\geq 1$ define

$$S_k(p_n)=p_n+p_{n+1}+\cdots+p_{n+k-1},$$

where $p_{n+i}$ are consecutive primes. Infinite admissible lengths conjecture. For every prime number $p_n$, there exist infinitely many odd integers $k\geq 3$ such that

$$S_k(p_n)$$

is a prime number. This strengthens the existence conjecture by asserting recurrence rather than merely one admissible length; the source supports it with numerical evidence and heuristic probabilistic arguments, but gives no proof.