Sharp prime gap conjecture $|\pi_2(p) - C_2 li_2(p)| < \frac{\sqrt{p+2}}{3}+2$

Let $\pi_2(n)$ be the prime twin counting function.

Let $li_2(x) = \int \frac{1}{\ln(t)}dx = li(x) - \frac{x}{\ln(x)}$

Let $C_2$ be the prime twin constant around $1.32$.

Then we get the sharp conjecture for a prime $p > 5$ :

$$|\pi_2(p) - C_2 li_2(p)| < \frac{\sqrt{p+2}}{3}+2$$

GENERALIZED CONJECTURE :

For prime gap of even distance $G$ :

Then we get the sharp conjecture $A$ for a prime $p > 5$ :

$$|\pi_G(p) - C_G li_G(p)| < \frac{\sqrt{p+G}}{3}+G$$

Where the LHS follows the Hardy-Littlewood notation.

I remark that no Skewes number for the " sexy primes "; primes with gap $6$, is known.

Ofcourse I am not asking for a proof. But some thoughts or counterexamples if they exist would be welcome.