Sick pay from health insurance while receiving treatment in another EU country

Dear community,

We are in a tight spot and we would love some help.

My wife is a full-time employee in Berlin and was diagnosed with cancer earlier this year. She informed her employer of this news and went on sick leave immediately. 

She is a citizen of another EU country so she decided to move there temporarily to receive the medical treatment in her hometown while getting the support from her family and friends. She told the doctors in Berlin that she was going to do this, but she has not informed TK, her public health insurance provider.

TK has been covering her sick pay until now based on the sick notes from her gynaecologist; however, they just reached out to us saying they need a report from the medical service (Gutachten des Medizinischen Diensts) by mid-Dec in order to assess my wife's incapacity to work. Otherwise, they will stop paying the sick pay.

My questions are:

1. Does my wife need to receive the medical treatment and/or be in Germany in order to receive the sick pay from TK? In other words, can my wife provide the medical report from her EU country and keep receiving the sick pay?
    
2. In case we informed TK that my wife has been receiving the treatment in another EU country, would that trigger TK to demand a refund of the sick pay they've paid so far? Or would TK simply stop paying the sick pay and not ask for a refund?
    

Thank you for reading this long message. Any advice or pointers would be highly appreciated! Happy to add any other information that might help.