He sustained a back injury from the fall and has been undergoing a period of convalescence.
It is possible that the "lost voice" narrative stems from social media rumors or confusion with other celebrities named Lambert (such as Adam Lambert, who was involved in recent media controversy). 2022---Christophe-Lambert--the-truth-behind-his-lost-voice
While your query mentions 2022, major news regarding Lambert's physical health surfaced more recently: He sustained a back injury from the fall