How much can an autocorrelation of a non-negative function resemble an indicator function?

The problem

Let $C$ be the best constant for which one has

$$\|f * f\|_{L^2(\mathbb{R})}^2 \leq C \|f*f\|_{L^1(\mathbb{R})} \|f * f\|_{L^\infty(\mathbb{R})}$$

for non-negative $f \colon \mathbb{R} \to \mathbb{R}$. What is $C$?

Statement quoted from problem 3 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve improved the best known bound; the record has since been surpassed. The problem remains open. Status in the repository: Former new result, since improved upon. See Section 6.2 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

Notes from the repository

Upper bounds for $C$

Lower bounds for $C$

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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