The Recursive Twin Pair Conjecture [Updated]

Let’s be (p,q)_k the k-th pair of twin prime, then for each twin prime p > 11, it exists a i-th, j-th pair of twin primes (p1,q1)_i, (p2,q2)_j where i < j < k like :

p ≔ p1 + p2 + 1

q ≔ q1 + q2 - 1

Example:
(p1,q1)_i≔(5,7)₂
(p2,q2)_j≔(11,13)₃
(p,q)_k≔(17,19)₄

p≔17≔ 5 + 11 + 1
q≔19≔ 7 + 13 - 1

Another visual example can be found underneath :

See also my previous post about: The New Prime Numbers Conjecture.

New Prime Numbers Conjecture (derivated from Goldbach's conjecture)

The Weak Form of the Prime Numbers Conjecture:
Every prime number p ≥ 5 can be written as the sum of two primes plus 1.

p_k ≔ p_i + p_j + 1, where k > j ≥ i

The Strong Form of the Prime Numbers Conjecture:
Every prime number p > 7 can be written as the sum of two distinctive primes plus 1.

p_k ≔ p_i + p_j + 1, where k > j > i and p_i ≠ p_j

Proof of the weak form:

According to Goldbach’s Conjecture:

“Every even number greater than 2 can be expressed as the sum of two primes.”

This statement is equivalent to say: “That every odd number greater than 3 can be expressed as the sum of two primes plus 1.” And, because a prime number is an odd number, this implies that every prime number greater or equal 5 can be expressed as the sum of two primes plus 1.
Therefore, in order to proof the weak form of the prime numbers conjecture, we have only to proof the Goldbach’s Conjecture.

Proof of the strong form:
(currently no proof for the strong form is available!)

Example:
3=1+1+1
5=1+2+1
    ***
7=3+3+1
    ***
11=3+7+1
13=5+7+1
17=5+11+1
19=5+13+1
23=5+17+1
29=11+17+1
31=13+17+1
37=5+31+1
41=3+37+1
43=13+29+1
...