While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense).

QUESTION. Does this hold true? If yes, how? $$\lim_{s\rightarrow\frac12^+}\sum_{n,m\geq1}\frac{2s-1}{n^sm^s(n+m)}=\pi.$$

Edited. $s\rightarrow\frac12$ has been replaced by $s\rightarrow\frac12^+$.