Gonek's negative moment conjecture for the Riemann zeta-function

Let $k>0$ be fixed, and let $T$ tend to infinity with $[?]$—the integrals below are taken along the line $[?]$. Gonek's conjecture. Uniformly for $1\leq\delta\leq\log T$,

$$\frac{1}{T}\int_0^T\left|\zeta\left(\frac{1}{2}+\frac{\delta}{\log T}+it\right)\right|^{-2k}\,dt\asymp\left(\frac{\log T}{\delta}\right)^{k^2},$$

while uniformly for $0<\delta\leq 1$,

$$\frac{1}{T}\int_0^T\left|\zeta\left(\frac{1}{2}+\frac{\delta}{\log T}+it\right)\right|^{-2k}\,dt\asymp\begin{cases}(\log T)^{k^2}&k<\frac12,\\(\log\frac{e}{\delta})(\log T)^{k^2}&k=\frac12,\\delta^{1-2k}(\log T)^{k^2}&k>\frac12.\end{cases}$$

This conjecture predicts the transition in the negative moments of the Riemann zeta-function as the shift approaches the critical line. It was proposed by Gonek; the supplied source does not state whether it has been proved or disproved.