Midpoint approximation for jumps of the Chebyshev psi function

Let $p_n$ be the $n$-th prime, let $\psi(x)=\sum_{m\leq x}\Lambda(m)$, and define

$$K_{n-1}=\operatorname{li}[\psi(p_n)]-\operatorname{li}[\psi(p_n-1)].$$

Midpoint jump conjecture. The jump at the prime $p_n$ satisfies

$$K_{n-1}=\widetilde K_{n-1}+O(1/p_n^2),\qquad \widetilde K_{n-1}=\frac{\log p_n}{\log\left[(\psi(p_n)+\psi(p_n-1))/2\right]}.$$

In particular, the sign of $\widetilde K_{n-1}-1$ is the sign of $K_{n-1}-1$. This approximation is used to control the sign of jumps and hence the Chebyshev-prime classification.