I've asked the same question on MathOverflow two days ago as On a conjecture about the arithmetic function that counts the number of twin primes, I add this reference while I hope to know what about my MathOverflow post.
In this post we denote (for a fixed positive integer or real number $x$) as $$\pi_2(x)=\#\{\text{ primes }p\leq x\,:\,p+2\text{ is also a prime}\}$$ the arithmetic function that counts the number of primes $p$ less than a given positive real $x$ satisfying that $p+2$ is also a prime. As general reference I add the article from Wikipedia Twin prime that refers that is unproven the existence of infinitely many twin primes, and the articles also from Wikipedia Second Hardy–Littlewood conjecture. I was inspired in these articles and a few experiments using Pari/GP scripts to state the following conjecture.
Conjecture 1. One has $$\pi_2(x+y)\leq \pi_2(x)+\pi_2(y)+1$$ for all integer $x\geq 2$ and all integer $y\geq 2$.
Conjecture 2. There exists an integer constant $K\geq 2$ such that $$\pi_2(x+y)\leq \pi_2(x)+\pi_2(y)+1$$ holds, for all integer $x\geq K$ and all integer $y\geq K$.
Question. Are these known, or is it possible to prove or refute any of previous conjectures? Can you find counterexamples for the first conjecture or add heuristics to know what about the veracity of this kind of conjectures? Many thanks.
I've tested the first conjecture for the segments of integers $2\leq x,y\leq 500$. I've added the second conjecture since I don't know if it is possible to find easily a counterexample for Conjecture 1.
Edit: Additionally you can investigate what about the veracity of the conjectures on assumption that the Twin Prime Conjecture holds. See the comments, many thanks to this user for this contribution.
References:
[1] G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. (44), J. E. (1923) pp. 1–70.