Asymptotic expression around $x=\infty$ of Infinite sum of the exponential integral: $\sum_{i=1,3,5..}\text{Ei}\left(\frac{-i^2 \pi^3}{4 x^2}\right)$

I ran across the following conjecture, which I checked numerically and seemed to check out. For $x\to \infty$ the sum$$\Theta(x)=\sum_{i=1,3,5..} \text{Ei}{\left(-\frac{i^2\pi^3}{4 x^2}\right)},$$has the asymptotic expression$$\Theta(x)_{x\to \infty}=\ln{2}-\frac{x}{\pi},$$where $\text{Ei}(x)$ is defined as$$\text{Ei}(x)=\int_{-\infty}^x\frac{e^t}{t}\text{d}t.$$

Showing that this is actually true proves to be a significantly hard task that once again, I'm asking for your help. How would one go about proving this?