lost recidence card outside Japan

I'd like to know if my mother can still go to Japan despite losing her Japanese residence card. She is currently staying here in the Philippines and is planning to go back to Japan in December.

google the answer with the words "在留カード 紛失 海外".

The only place where resident cards can be re-issued is in Japan.

As far as I know:
If she has left Japan with re-entry permit, then she should be able to enter Japan again with her passport, as immigration authorities should have a record of her departure from Japan with re-entry permit.

But since the Philippines is not a visa waiver country with Japan, just to ensure that she can leave the Philippines for Japan (as a proof to the airline that she does have a visa for Japan), she might have to go through this process:
https://www.moj.go.jp/isa/content/930002837.pdf
She first needs to report to the local police that she lost it (get a police certificate),
Get the certificate translated into Japanese,
Ask a friend/relative, etc. in Japan to apply for the above with the immigration bureau in Japan (power of attorney is needed for that person to do it on her behalf), then have them mail it to her in the Philippines.
With this certificate, she should be allowed onboard an aircraft to Japan, and enter Japan. (Then she needs to apply for a new resident card within 14 days of arriving in Japan.)