Primes in almost all short intervals III

A conjecture states that if the interval $[n-n^{\theta_{0}+\varepsilon}, n]$ contains primes for all $n$, then the interval $[n-n^{\theta_{1}+\varepsilon}, n]$ contains primes for almost all $n$, where $\theta_1 = 2 \theta_0 - 1$. In the present paper, the author proves that $\theta_1 = \frac{1}{24}$, improving the previous exponents $\frac{1}{21.5}$ and $\frac{1}{22}$ by the author himself. If the conjecture is true, then we can get $\theta_1 = \frac{1}{25}$.