On the quantity of twin prime pairs of a given form

Question

Let $p_l$ the $l$-th prime number. I've considered the formula

$$\frac{N_{n+1}}{N_n}+\frac{N_{n+2}}{p_{n+1}N_n}\pm1$$ where $N_k=\prod_{l=1}^k p_l$ is the primorial of order $k$. Previous formula can be simplified thus as $$p_{n+1}+p_{n+2}\pm 1.\tag{1}$$

I wondered, for $n\geq 1$ running over positive integers, when the terms of previous sequence provide us twin prime pairs $$(p_{n+1}+p_{n+2}-1, p_{n+1}+p_{n+2}+1).\tag{2}$$

Question. Let $p_n$ be the $n$-th prime number. Is it possible to prove that there exist finitely many twin prime pairs of the form $$(p_{n+1}+p_{n+2}-1, p_{n+1}+p_{n+2}+1)$$ when $n$ runs over positive integers ? Many thanks.

The last paragraph of the section Properties from the Wikipedia Twin prime tell us the elementary sieve that satisfy the sequence of twin primes. I don't know if it, or Brun's theorem, can be useul.

Computational evidence. 1) My belief is that the formula $(1)$ provide us many twin prime pairs, but I don't know if it is a remarkable proportion of those. You can evaluate the code (it is a line) using the web Sage Cell Server (choose GP as language)

for(n=1, 100, if(isprime(prime(n+1)+prime(n+2)-1)==1&&isprime(prime(n+1)+prime(n+2)+1)==1,print(prime(n+1)+prime(n+2)-1)))

And you can do a comparison using this other code

for(n=1, 881, if(isprime(n)==1&&isprime(n+2)==1,print(n)))

Previous outputs are thus terms of the sequence A001359, from the OEIS. You can to evaluate the code for different segments of integers $1\leq n\leq N$, and choose a different output from the function print()

2) My belief is that this sequence/s aren't in the literature.

Answer 1

Indeed, standard conjectures imply that there should be infinitely many such twin prime pairs even if we insist that $p_{n+1},p_{n+2}$ are themselves twin primes! In other words, there should be infinitely many integers $p$ such that $p$, $p+2$, $p+(p+2)-1=2p+1$, and $p+(p+2)+1=2p+3$ are all prime—this is a special case of Schinzel's Hypothesis H (and almost a special case of the Hardy–Littlewood conjectures prime $k$-tuples conjecture).