In this postthis post 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:
$$\dfrac{\Psi}{\Gamma}(s) \pm \dfrac{\Psi}{\Gamma}(1-s)$$
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)$$\zeta(s) \pm \zeta(1-s)$ that (assuming the RH) also induces a finite few complex zeros lying off the critical line and these also seem to be absent when using:
$$\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$$
as framed up in this questionthis question.
