Citing "In press" article I had a conference last week on which I presented paper 1.

There's paper 2 that has been accepted with the CR deadline within a week. I'd like to cite paper 1 in paper 2. I could make the reference state "In press" but my worry is, when paper 2 gets published, is the "In press" part going to remain there after the publication (forever)?

Is there a way to avoid this or solve this?
Many published articles cite "In press" articles forever and it's not an issue.

You could, however, publish a preprint (on arXiv, on PeerJ, on ResearchGate, etc). This will allow you to have a DOI for the unpublished article. You need to be very careful: not all journals allow preprints (I do not know about conferences), so it could jeopardize the publication of paper 1. You have to verify your signed agreement (if you have one) or ask.