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).

Is the sunflower conjecture true?

This question will resolve positively in the event of a publication in a major mathematics journal proving the sunflower conjecture. It will resolve negatively in the event of a publication in a major mathematics journal disproving the sunflower conjecture.

If there is no such proof by 2300-01-01, the question will resolve ambiguous. If a proof is published, but not confirmed by peer review by 2300-01-01, the question may wait to resolve until peer review has reached a consensus.

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[EDIT] Sylvain 2021-09-08 : changed the resolution date from 2121 to 2300.

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