Conjecture on absolute moments of primes in relative short intervals

Let $X$ be large, let $\delta$ 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. Relative short-interval moment conjecture. For $X^{-1+\epsilon}\leq\delta\leq X^{-\epsilon}$,

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

This is the multiplicative-interval analogue of the preceding prediction and extends the established moment asymptotics beyond the integer moments treated in the paper; the conjecture is presented as an expectation and no resolution is given.