If proven, would the solution to this integral imply a zero free region or strip in the critical strip?

I have found the following relationship starting from the Volchkov integral here https://mathoverflow.net/a/57944/25104, where the number $+4$ is only a conjecture yet shown numerically to about $8$ or $9$ decimal places depending on the choice of epsilon $\epsilon$

$\underbrace{\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{\frac{1}{2}+\epsilon}^{\infty}\Re\left(\left.\int\frac{\zeta'(s)}{\zeta(s)}~ds\right|_{s=\sigma+it}\right)~d\sigma ~dt}_{\text{The Volchkov integral with a small number epsilon }\epsilon} = \\ \lim_{s \rightarrow \epsilon} \left(\frac{1}{32} \pi \left(\underbrace{-\frac{\zeta '(s+1)}{\zeta (s+1)}+\frac{1}{(s-1) s}}_{\text{Theorem/easily explainable}}\underbrace{+4}_{\text{Conjecture/Riemann hypothesis}}\right)\right)$

By changing the epsilons $\epsilon$ into $\epsilon_1$ and $\epsilon_2$, like this:

$\underbrace{\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{\frac{1}{2}+\epsilon_1}^{\infty}\Re\left(\left.\int\frac{\zeta'(s)}{\zeta(s)}~ds\right|_{s=\sigma+it}\right)~d\sigma ~dt}_{\text{The Volchkov integral with a small number epsilon }\epsilon_1}-\underbrace{\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{\frac{1}{2}+\epsilon_2}^{\infty}\Re\left(\left.\int\frac{\zeta'(s)}{\zeta(s)}~ds\right|_{s=\sigma+it}\right)~d\sigma ~dt}_{\text{The Volchkov integral with a small number epsilon }\epsilon_2} = \\ \lim_{s \rightarrow \epsilon_1} \left(\frac{1}{32} \pi \left(\underbrace{-\frac{\zeta '(s+1)}{\zeta (s+1)}+\frac{1}{(s-1) s}}_{\text{Theorem/easily explainable}}\underbrace{+4}_{\text{Conjecture/Riemann hypothesis}}\right)\right)-\lim_{s \rightarrow \epsilon_2} \left(\frac{1}{32} \pi \left(\underbrace{-\frac{\zeta '(s+1)}{\zeta (s+1)}+\frac{1}{(s-1) s}}_{\text{Theorem/easily explainable}}\underbrace{+4}_{\text{Conjecture/Riemann hypothesis}}\right)\right)$

could one somehow by complex analysis get rid of the conjectured unknown quantity represented by the number $+4$ in the limits? Meaning $4-4=0$ or similar?

And would it imply a zero free region for Riemann zeta between $\Re(s)=\frac{1}{2}+\epsilon_1$ and $\Re(s)=\frac{1}{2}+\epsilon_2$?