Are both stems part of the same root crown? If not, the one with no buds hasn't rotted yet. If this is one plant, I would wait until all green leaves the stem before I'd cut it. Often, plants don't grow symmetrically, and it's worth a shot to see if it's just slow. Of course, when the stem is completely dead, or the lower portion of it is, it's done for in that position and you can cut it off.
The buds on the other stem look healthy and promising. They make me think that side is beginning ti root in. If you want, you can check the base of the other stem. If it's dry/dead at the bottom, you could remove the dead when cutting it off, then replant the remaining section (I see green) as a cutting.