Midpoint approximation for jumps at prime powers

Let $p_n$ be the $n$-th prime, let $l\geq1$, and define the jump at $p_n^l$ by

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

Prime-power midpoint conjecture. The jump satisfies

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

In particular, the sign of $\widetilde K_{n-1}^{(l)}-1$ is the sign of $K_{n-1}^{(l)}-1$. This is intended to approximate the jumps defining generalized Chebyshev primes.