Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?

Question

I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:

For $n, x \ge 2$ be two integers then: $$P_{2n} \ge 2P_n$$ and $$\pi(2x) \le 2\pi(x)$$

Where $P_n$ is $n$-$th$ prime and $\pi(x)$ is Prime-counting function

My question: Is the conjecture above contradictory (inconsistent) to the Prime k-tuple conjecture? I am looking for a proof, comment or reference.

Answer 1

We have $\pi(2x)<2\pi(x)$ for any $x\geq 11$. For a proof see here. Similarly, it follows from standard bounds that $P_{2n}>2P_n$ when $n$ is sufficiently large.