Non-EU citizen working as salaried PhD in Denmark

Dear Sir/Madam,

Currently I am a full-time employed phd student in Aarhus university. I have a questions about my eligible to get a unemployment fund during the jobseeking period. Here are the some informations about my situation.

1- I got accepted phd in September 2019. I am working as a phd until then. They gave me visa for 3.5 years. 3 years for my phd then 6 months for the jobseeking.

2- I will finish my phd in August 2022. Then I have additional 6 months of residence permit for jobseeking.

3- I work 37 hours per week for now. However after my phd is done, I can work 20 hours in the jobseeking period they gave me (from august 2022-february 2023).

4- It is also says I cannot continue to work full-time in other employment unless

I apply for an extension of my residence permit.

5- I have not registered as a-kasse member until now.

I am wondering; if I start paying for a-kasse – I still have more than 1 year until my employment finish at Aarhus university- can I get unemployment benefits for my jobseeking period ? I understand that to get the benefits you have to have rights to start a job one day notice but in my visa it says I may work 20 hours in my jobseeking period. So should I start paying for a-kasse to get benefit in my 6 months jobseeking period ?