MySQL 8.0

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create table t (js json);
insert into t values('{"slug": "SLUG","price": {"44":12,"_default":12}, "test": "TEST"}');
update t
set js = json_replace(js, '$.price', json_object('44', 13, '_default', 13))

Rows matched: 1  Changed: 1  Warnings: 0

select * from t

js

7b22736c7567223a2022534c5547222c202274657374223a202254455354222c20227072696365223a207b223434223a2031332c20225f64656661756c74223a2031337d7d