Should the wife continue her marital relationship with her husband and conceal her previous adultery from him, or should she inform him, and what is the ruling on their relationship after the husband has repeatedly made the pronouncement of her triple divorce contingent upon proof of her adultery before marriage?

First, you must repent to Allah for the sin committed. The fatwa (definition) states that a triple divorce pronounced at once counts as three, and a conditional divorce takes effect when its condition is met. Accordingly, you have been divorced from your husband three times, and a major irrevocable divorce (baynunah kubra) has occurred between you. You must separate from him. It is advisable to present the matter to the Sharia (definition) court for their ruling.