Negative-jump conjecture for the truncated psi-based prime-counting function

Let $\mu$ be the Möbius function, let $\psi(x)=\sum_{m\leq x}\Lambda(m)$, and for an integer $N>1$ define

$$\eta_N(x)=\sum_{n=1}^N\frac{\mu(n)}{n}\operatorname{li}[\psi(x)^{1/n}]-\pi(x).$$

Negative-jump conjecture. Negative jumps of $\eta_N(x)$ occur at every point $x+1\in\mathcal P$, where $\mathcal P$ is the set of primes, and

$$\lim_{p\to\infty}[\eta_N(p)-\eta_N(p-1)]=0.$$

For $N>1$, this is based on numerical observations that the negative jumps occur at all relevant primes and decrease in amplitude toward zero.