Is the sunflower conjecture true?

One of Paul Erdős' favorite problems was the sunflower conjecture, due to him and Rado. Erdős offered $1000 for its proof or disproof.

The sunflower problem asks how many sets of some size (n) are necessary before there are some (3) whose pairwise intersections are all the same. The best known bound was improved in 2019 to something the form ( \log(n)^{n(1+o(1))} ); see here for the original paper and here for a slightly better bound. The sunflower conjecture asks whether there is a bound (c^n) for some constant (c).