An identity for the Lambert $W$ function

Question

Expressing the integral in An integral identity in terms of residues, we come to the following supposed identity: $$\sum_{k=-\infty}^\infty\frac1{1 + W_k(x)}=\frac12$$ for all $x\in(-1/e,0)$, where $W_k$ is the $k$th branch of the Lambert $W$ function.

How can this be proved?

Answer 1

This identity is now established, since An integral identity cited in the above post is now proved.