Odd-sunflower exponential bound conjecture

A $k$-uniform family is a family of sets in which every set has size $k$. An odd-sunflower is a family of at least two sets such that every element of the underlying set belongs to an odd number of the sets or to none of them. Odd-sunflower exponential bound conjecture. The maximum size of any odd-sunflower-free $k$-uniform family is at most $c^k$ for a suitable constant $c>0$. This is posed as a weakening of the Erdős–Rado sunflower conjecture and is left open in the paper.