Microscopic zeta logarithmic-derivative moment conjecture

Let $\omega$ be uniform on $[0,1]$, let $T\to\infty$, and let $a,a'\in\mathbb C$. The source considers the logarithmic derivative of the Riemann zeta function at points shifted from the critical line by $a/\log T$ and $a'/\log T$.

Microscopic logarithmic-derivative moment conjecture. The following two asymptotic identities should hold:

$$\lim_{T\to\infty}\frac{1}{\log^2T}\mathbb E\left[\frac{\zeta'}{\zeta}\left(\frac12+i\omega T+\frac{a}{\log T}\right)\frac{\zeta'}{\zeta}\left(\frac12+i\omega T+\frac{a'}{\log T}\right)\right] =\mathds{1}_{\operatorname{Re}(a)<0,\operatorname{Re}(a')<0}-\frac{1-e^{-(a'-a)\operatorname{sgn}\operatorname{Re}(a'-a)}}{(a-a')^2}\mathds{1}_{\operatorname{Re}(a)\operatorname{Re}(a')<0},$$ $$\lim_{T\to\infty}\frac{1}{\log^2T}\mathbb E\left[\frac{\zeta'}{\zeta}\left(\frac12+i\omega T+\frac{a}{\log T}\right)\overline{\frac{\zeta'}{\zeta}\left(\frac12+i\omega T+\frac{a'}{\log T}\right)}\right] =\mathds{1}_{\operatorname{Re}(a)<0,\operatorname{Re}(a')<0}+\frac{1-e^{-(a+\overline{a'})\operatorname{sgn}\operatorname{Re}(a+\overline{a'})}}{(a+\overline{a'})^2}\mathds{1}_{\operatorname{Re}(a)\operatorname{Re}(a')>0}.$$

These formulas are proposed as consequences of the preceding process conjecture, provided the relevant moments are controlled. They describe second moments on the microscopic scale and remain conjectural.