A limit arising from a rigidity problem for linear differential equations

I am studying a family of non‑homogeneous linear complex differential equations and encountered the following limit. I would like an explicit counterexample, if one exists.

We consider $\eta \in L^\infty([1,+\infty))$ satisfying the following hypothesis $(H)$:

$$ \exists \rho_\eta \in \mathbb{C}^* \quad \text{such that} \quad \sup_{t \ge 1} \left| \int_1^t \big( \eta(u) - \rho_\eta \big)\, du \right| < +\infty. $$

For $w\in\mathbb{C}_+:=\{z\in\mathbb{C}:\Re(z)>0\}$ define $$\mu_\eta(w) = -1-(1-w)\int_1^{+\infty} u^{-1-w}\,\eta(u)\,du .$$ (The integral converges absolutely because $\Re(w)>0$ and $\eta$ is bounded.)

Let $B:=\bigl\{w\in\mathbb{C}:\Re(w)\in(0,1),\ \Re(w)\neq\frac12,\ \Im(w)>0\bigr\}$.

In a previous paper (see reference below) the following was observed as a remark:

Assume $\eta$ satisfies (H). Let $s=\sigma+i\tau\in B$ and suppose $\mu_\eta(1+i\tau)\neq0$. If $\mu_\eta(s)=0$ or $\mu_\eta(1-\overline{s})=0$ (it is known that they cannot both be zero), then the following limit should be $+\infty$: $$L(s,\eta) := (2\sigma-1)\lim_{t\to+\infty}\Re\!\left(\frac{\mu_\eta(s)\,t^{\sigma} - \mu_\eta(1-\overline{s})\,t^{1-\sigma}}{\mu_\eta(1+i\tau)}\right) = +\infty . \tag{1}$$

Question: Is this statement always true under the stated hypotheses? If not, can you provide an explicit counterexample with:

• a function $\eta$ satisfying (H),
• $s\in B$ with $\mu_\eta(1+i\tau)\neq0$,
• $\mu_\eta(s)=0$ or $\mu_\eta(1-\overline{s})=0$ (exactly one of them),
• but for which $L(s,\eta)$ is not $+\infty$.

What I have tried so far:
I performed numerical experiments with families like $\eta(u)=1+c\,\frac{\cos(\ln u)}{u}$ (choosing $c$ to make one of $\mu_\eta(s),\mu_\eta(1-\overline{s})$ zero). In all tested cases with $s=0.6+2i$, I obtained $L(s,\eta)=+\infty$. Combinations of several frequencies gave the same result. Trivial $\eta\equiv1$ does not satisfy the vanishing condition. So far no counterexample has been found, suggesting the statement might be true, but I have no rigorous proof.

Any help (remarks or counterexample) is welcome.

Reference: W. Oukil, *Rigidity and Structural Asymmetry of Bounded Solutions*