Expression for the causal retarded potential for $t<0$ must give $0$ but my calculation produces a nonzero result. What's the mistake?

This question was previously asked here in the Mathematics StackExchange but using a slightly different notation. But I did not find the answer I was looking for or rather got two very different answers.

Consider the following integral $$I=intlimits_{-infty}^{infty}domega frac{e^{-iomega t}}{omega^2-c^2k^2}$$ which has poles at $omega=pm ck$. When poles exists on the real line, we can either indent the contour to bypass the poles (or equivalently, use an appropriate $ivarepsilon$-prescription).

Let us choose a contour that lies entirely in the upper half of the complex $omega$ plane. Let the contour consists of two small C1 and C2, each of radius $varepsilon$, that goes over the poles at $omega=pm ck$, and the larger semicircle $C$ also lies in the upper half of the complex plane. Please see the diagram of Tong’s Lecture notes on Electromagnetism on page 125, figure 50.

With this choice of contour, I Tong shows that $I=0$ for $t<0$.

But when I am trying to calculate it, for different parts of the contour, piece by piece, I get a nonzero result. Let me discuss how. Using Cauchy’s residue theorem and Jordan’s lemma, gives (schematically),

$$0=intlimits_{-infty}^{-ck-varepsilon} +intlimits_{-ck-varepsilon}^{-ck+varepsilon}+intlimits_{-ck+varepsilon}^{ck-varepsilon}+intlimits_{ck-varepsilon}^{ck+varepsilon}
+intlimits_{ck+varepsilon}^{+infty}.$$

In the limit, $varepsilonto 0$, the sum of the 1st, 3rd and 5th integrals reduce to the required integral $I$. Thus, $$I+intlimits_{-ck-varepsilon}^{-ck+varepsilon}+intlimits_{ck-varepsilon}^{ck+varepsilon}=0.$$ Therefore, in the limit $varepsilonto 0$, $$I=ipi{rm Res}(z=+ck)+ipi{rm Res}(z=+ck)
=ipileft(frac{e^{-ickt}}{2ck}right)+ipileft(frac{e^{ickt}}{-2ck}right)
=frac{pi}{ck}sin(ckt)neq 0$$

Therefore, this does not agree with Tong’s notes I linked. What is wrong with my calculation? Tong’s result is correct but I am unable to reproduce it in my way. Please note that I am not directly using the $iepsilon$-presciption of shifting poles but using the prescription of indenting contours.