Got put in a PIP in a japanese company. (in IT and I need advice)

Working for a Japanese company rn as an IT engineer and got put on PIP because of delays in my module(i got assigned a huge module to develop from scratch and had delays with the deliverables about a week or 2). They asked me to undertake PIP which, they said, if I fail, will lower my salary. If I don't take the PIP they'll give me the pay cut eitherway. There was no mention of termination. Overall, my work environment and take home pay is good.
PIP ends february and we're at the action plan/goal setting stage. I think
Should I start looking for a job now? Is it possible that they're lying and they're just trying to fire me? Or should I finish my PIP?

I have an N3 and 10y work exp in software dev but I'm worried I won't find another company that would accept me that's full remote. I only have an engineer visa valid for 3 more years.
I'm worried if I don't go find a job soon I'll get kicked out of Jp.

Edit: Thank you for your replies. I'm a 正社員 btw.