Is the wife required to wear the hijab in front of her unbelieving father-in-law, who is not married to her husband's mother, and her husband was born out of wedlock?

The majority of jurists hold that a child born out of wedlock is not attributed to the fornicator, meaning there is no prohibitive degree of kinship (mahramiyyah) between them, nor is there inheritance. Some scholars, such as Al-Hasan, Ibn Sirin, and Ibn Taymiyyah, are of the view that the child is attributed to the fornicator if he claims him and attributes him to himself, and consequently, inheritance and mahramiyyah ensue. If the fornicator attributes his son to himself, then he is his son according to the second opinion, and he would be a mahram to you. However, it is more cautious for you to wear the hijab in front of him, out of consideration for the opinion of the majority.