I'm not sure I get your first question - my post set out what I believe the conflict is - not meeting the criteria to represent her in a death penalty case. That is definitely a conflict, in my opinion. It might not be right, but it is my guess.
MM filed a notice of appearance in the murder case on 27th May, the day after her continued initial appearance on the murder charges. Perhaps because of her indigent status, I don't know, the State had an interest in asking if he would be entering a notice of appearance.
https://coi.isc.idaho.gov/docs/case/CR22211624/Notice of Appearance Reservation of Defenses.pdf
The State has filed its objection, it looks to me, once it had gone through the process of deciding to seek the death penalty and perhaps filed its objection at that time because that's when his appearance became conflicted (in view of the stay), so that the court would have time to appoint qualified counsel.
JP is privately retained so he can represent Chad in a death penalty trial if that's what Chad wants and can finance it himself. It's a different matter when/if Chad becomes indigent and if JP will resign from the case or become co-counsel. Chad was asked if he wanted to represent himself, or hire JP, or seek application of a public defender at his initial appearance.
MOO