Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?

Question

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Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of:

$$Li_s(z)\,- \, Li_{1-s}(z)$$

all converge towards the shape $s=\dfrac12 + \dfrac{k \, \pi}{\,\ln(2)}i\, $ with $k \in \mathbb{Z}$.

Interested to understand whether this apparently trivial result could be derived from any known formulae for $Li_s(z)$ like:

$$Li_s(z) = \frac{\Gamma(1-s)}{(2\,\pi)^{1-s}} \left(i^{1-s}\,\zeta_H\left(1-s,\frac12+\frac{\ln(-z)}{2 \,\pi \, i}\right)+i^{s-1}\,\zeta_H\left(1-s,\frac12-\frac{\ln(-z)}{2 \,\pi \, i}\right)\right)$$

Thanks.