Most likely the conservatorship will control. You may want to consult an attorney and bring up these issues in court-- the may appoint a court appointed guardian ad litem to advocate on behalf of you sister.
If there is a conservatorship, and if your sister is unable to express her preferences, then the conservatorship will take precedence. A Power of Attorney is designed to give the agent or "attorney-in-fact" the same rights and powers a Conservator would have, but without the court proceedings. Once the probate court becomes involved, the Power of Attorney would be superseded.
Once a petition for conservatorship is filed, the court will appoint a Guardian ad litem, (GAL) to investigate and make a report to the judge. It is likely, under your facts, that if there is no agreement on who should act, that the court will appoint a public administrator to act on behalf of your sister. Obviously, if a petition is filed, you are going to need to object to it.
Unfortunately, it would appear that you are going to need to fight this in court.
Best of luck to you!
Many times the Court will consider who holds the power of attorney at the hearing for the for the guardian/conservatorship as evidence or guidance of what the ward would want. However, the Court will give great weight to the GAL's opinion in making their appointment. Consulting an attorney will help you weigh these considerations.