Near-critical logarithmic-derivative moment conjecture for the Riemann zeta-function

Let $K$ be a positive integer, let $a>0$, and consider the logarithmic derivative of the Riemann zeta-function at $1/2+a/\log T+it$. Near-critical logarithmic-derivative moment conjecture. There is a function $c(a)$ such that

$$\lim_{T\to\infty}\frac{1}{T(\log T)^{2K}}\int_0^T\left|\frac{\zeta'}{\zeta}\left(\frac12+\frac{a}{\log T}+it\right)\right|^{2K}\,dt=c(a),$$

with

$$\lim_{a\to0^+}c(a)(2a)^{2K-1}=\binom{2K-2}{K-1}.$$

This conjecture gives the zeta-function analogue of the paper's random-unitary-matrix asymptotic for logarithmic-derivative moments near the unit circle. The source does not state a resolution.