Mixed logarithmic-derivative moments conjecture for the Riemann zeta-function

Let $Z(t)$ be Hardy's function, so that $|Z(t)|=|\zeta(1/2+it)|$, and let $\sigma_{I\!I\!I'}$ be the function defined by the Painlevé equation referenced in the source. For non-negative integers $K$ and $M$ with $M\le K$, the mixed-moments conjecture asserts that, as $T\to\infty$,

$$\frac{\displaystyle\int_0^T \left|\frac{Z'}{Z}(t)\right|^{2K-2M}|Z(t)|^{2K}\,dt}{\displaystyle\int_0^T |Z(t)|^{2K}\,dt} \sim \left(i\log T\frac{d}{dx}\right)^{2K-2M}\left.\exp\left(\frac{x}{2}-\int_0^{4x}\frac{\sigma_{I\!I\!I'}(s)+K^2}{s}\,ds\right)\right|_{x=0}.$$

This conjecture predicts mixed moments involving Hardy's function and its logarithmic derivative, extending the conjectural moment formulas for the Riemann zeta-function and the random-matrix analogues proved in the paper. Its resolution is not supplied in the source.