OP has the same ear as me. I agree that the AKM based Hegel DACs have a midrange dip, the H590 more so than the H390. I think this is why the H390 is perceived as warmer. I recently had the chance to A/B a H590 and new H600.
What I found was the new 600 DAC (ESS based I believe) has a much different presentation. The H600 is much more focused and has a more tube like feel. The vocals and especially instruments are more tightly concentrated. This does have a cost, vs. the H590, the H600s sound stage is not as wide and not nearly as tall. It seems that the H600 has sorted out the midrange dip.
If the OP likes the warmer sound maybe wait for the new Hegel DAC to trickle down into the H390? Good thread, cheers