Your conjecture is not compatible with some actual heuristic views:

**Cramer Conjecture:**
$$\limsup_{n\to+\infty}\dfrac{p_{n+1}-p_n}{\log(p_n)^2}=1$$

Then if this conjecture holds, we have infinitly many intervals of size $(1+o(1))\log(n)^2$ does not contain any prime numbers.

**Granvile conjecture:**
$$\limsup_{n\to+\infty}\dfrac{p_{n+1}-p_n}{\log(p_n)^2}\gtrsim2e^{-\gamma}\approx1.12$$
*($f(x) \gtrsim g(x) \iff f(x) \geq (1+o(1))g(x)$)*

Then if Granvile's conjecture holds, we have infinitly many intervals of size $(2e^{-\gamma}+o(1))\log(n)^2$ does not contain any prime numbers.

You can see that $2e^{-\gamma} > 1$, then **Granvile's conjecture holds** implies that your conjecture is false.