Conjecture on absolute moments of primes in short intervals

Let $X$ be large, let $h$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define $E=2\pi e^{C_0-1}$, where $C_0$ is Euler's constant. Absolute-moment conjecture. For $X^{\epsilon}\leq h\leq X^{1-\epsilon}$,

$$\int_{1}^{X}|\psi(x+h)-\psi(x)-h|^{\lambda}\,dx=\frac{\Gamma(\lambda+1)}{\Gamma\left({\lambda\over2}+1\right)2^{\lambda/2}}h^{\lambda/2+1}\int_{E}^{X/h}\left(\log{x\over E}\right)^{\lambda/2}\,dx+o(Xh^{\lambda/2}).$$

This extends the first main terms proved in the paper's theorems to arbitrary positive moments and predicts the full asymptotic uniformly throughout the indicated short-interval range; its resolution is not supplied here.