I have changed the way current is transferred to the moving wire. In my previous setup I used a carbon brush, which seemed like a nice idea, but the resistance of the brush turned out to be too high (1 ohm) which would introduce large losses and heat dissipation.
Instead, I now use a piece of HSS CO-8 very hard steel from a broken mill bit that I saved after a milling accident (never throw those away, they always come in handy!). The EDM wire drags over the steel making contact. The power is connected to the HSS piece by a silver soldered brass nut. I’ve seen a similar solution in Jaakko Fagerlund’s machine and Mariano also mentioned this solution in a comment on a blog post.