Flying through FRA and on to SIN next week. Fully vaccinated, but tested positive 2 weeks before my scheduled flight (and have PCR recovery cert). (I am now testing negative thankfully)

Would be really grateful if someone could confirm their understanding as follows:-
- I don't need to complete the online digital registration for Germany, as only transit
- I tested positive a couple of weeks ago, but proof of immunity (positive PCR) is only valid 28 days afterwards in Germany, and therefore...
- I still need to take an ART or PCR prior to the flight to Germany, even though my final destination SIN doesn't require it?

thanks in advance!