Does a divorce take effect if the wife divorces herself, knowing that the husband did not intend to authorize her to divorce when he said to her: "You say it"?

The wife is at fault for seeking divorce without a valid reason, based on the hadith: "Any woman who asks her husband for divorce without a sound reason, the fragrance of Paradise will be forbidden to her." And the husband's statement: "You say it," does not necessitate a divorce, because there was no intention to enact it, and intention is one of the pillars for divorce to occur. The wife's statement: "I am divorced," is not valid without authorization or empowerment from the husband.