Beautiful!
If all the components are stock (and they look so), that Deore mountain RD should take 34T no problem--though if a B-screw exists on this generation of Deore (I don't recall seeing one as the anchor pivot was non-sprung), it may likely need winding out. Even an earlier 600 Arabesque GS/long cage RD was spec'd to handle 34T--I've done it and it works very well. The RD on this bike looks a little worse for the non-wear. :/ Maybe some Tri-Flow will help.
Just checked--34T max cog is spec'd.