I know that the closest proven statement to the Twin Prime Conjecture is Chen's theorem, stating that there is an infinite number of primes $p$ such that $p+2$ is either prime or semiprime. The proof is however highly complicated; I was wondering if any of the following weaker statements have simpler or clever proofs:
Existence of an infinite number of primes $p$ such that $p+2$ has a bounded number of prime factors (either existence of bound or explicit bound)
Existence of an infinite number of primes $p$ and existence of a number $n$ (explicit or not) such that $p+n$ has a bounded number of prime factors
Existence of an infinite number of positive integers $k$ such that both $k$ and $k+2$ have a bounded number of prime factors
Existence of an infinite number of positive integers $k$ and of a positive integer $n$ such that both $k$ and $k+n$ have a bounded number of prime factors
By number of prime factors I guess that both number of $distinct$ prime factors and number of $overall$ prime factors in the factorisation are alright (although bounded number of overall factors implies bounded number of distinct factors so it suffices).