Hello toivo 1,
It does not look like a stye. It seems to be a chalazion. Most of the time a chalazion does not go with conservative management. A very minor surgical procedure called incision and curretage is done and is usually curative. Chalazion is painless whereas stye is usually painful and does not last for this lonh. You can wait for a few weeks to see whether it goes on this treatment or not and then can go for incision and curretage. It won't cause harm. Sometimes, it may go but is a little unlikely. Thanks.
Oral antibiotic is not needed. Just do warm compresses and a topical antibiotic eye ointment like tetracycline at bed time. Thanks.