Should I ask for equity when joining a startup? When/how? (And whom should I call via Clarity?)

A startup enrolled in accelerator (very prestigious) wants me to join them, after briefly trying me. The "deal" now is for 1 month fulltime and then we'll see – they'll cover costs of living, but otherwise almost no cash. They cannot give me equity now because of the accelerator restrictions. It's so interesting that I'd do it even for free, and I don't wanna be greedy. On the other hand, I don't want to regret later that I didn't ask for something I should have. My role is kinda crucial – being the main and only designer in the team. Not sure how to handle this, zero experience. And not sure whom to ask / call. Urgent of course :-) Thanks very much for guidance.