In [this post][1] it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.

I have been searching for similar reflexive $\pm$ $\Gamma$-functions that would not have a finite few exceptions and based on numerical evidence I like to conjecture that all zeros of: 

$$\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$$

or expressed differently with $\Psi(s)$ the [Digamma function][2]:

$$\dfrac{\Psi}{\Gamma}(s) \pm \dfrac{\Psi}{\Gamma}(1-s)$$

are either real or reside on the line with $\Re(s)=\frac12$.

**Question:**

Could this be proven using the techniques applied to $\Gamma(s) \pm \Gamma(1-s)$ ?

*Just as a side observation:*

A similar effect appears to occur for [$\zeta(s) \pm \zeta(1-s)$][3] that (assuming the RH) also induces a finite few complex zeros lying off the critical line and these also seem to be absent when using: 


$$\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$$

as framed up in [this question][4]. 


  [1]: https://mathoverflow.net/questions/89324/are-all-zeros-of-gammas-pm-gamma1-s-on-a-line-with-real-part-frac12
  [2]: http://mathworld.wolfram.com/DigammaFunction.html
  [3]: https://mathoverflow.net/questions/89518/are-the-semi-trivial-zeros-of-zetas-pm-zeta1-s-all-on-the-critical-li
  [4]: https://mathoverflow.net/questions/206198/are-all-complex-zeros-of-dfrac-zeta-zetas-pm-dfrac-zeta-zeta1-s