The sum-integral conjecture for the gamma-function logarithm

Let $z_m$ be the sequence of complex numbers used in the preceding gamma-function expression, and let $A\in\mathbb{N}$. The logarithm is taken on the branch denoted by $\operatorname{Log}$. Sum-integral conjecture. For any $A\in\mathbb{N}$,

$$\lim\limits_{A\to\infty} \sum\limits_{m=-A}^A \operatorname{Log}\left(\frac{\Gamma\left(1+A+i\overline{z_m}\right)\Gamma\left(1+A-i \overline{z_m}\right)}{\Gamma\left(1+A+iz_m\right)\Gamma \left(1+A-iz_m\right)}\right) = \lim\limits_{A\to\infty} \int\limits_{-A}^A \operatorname{Log}\left(\frac{\Gamma\left(1+A+i\overline{z_m}\right)\Gamma\left(1+A-i \overline{z_m}\right)}{\Gamma\left(1+A+iz_m\right)\Gamma \left(1+A-iz_m\right)}\right)\,dm.$$

This conjectured replacement of the finite sum by an integral is used to analyze the gamma-function contribution to the Hall conductivity of a chiral $p\pm ip$ superconductor; the supplied text gives no evidence that the assertion has been proved or disproved.