Am I obligated to try to appease my mother-in-law, knowing that my husband refuses to visit her or reconcile with her due to her false accusations against me, and he believes he has not wronged her?

The husband must repent to Allah for refusing to invite his mother, apologize to her, and endure her ill-treatment; for Paradise is at her feet. It is not permissible for children to deny the rights of their parents, even if they mistreat them, and they must show piety towards them, even if they are disbelievers.