Erdős's density conjecture for distance-1-avoiding sets in the plane

In the normed vector space $(\mathbb{R}^n,\|\cdot\|)$, a measurable set $A\subseteq\mathbb{R}^n$ avoids distance 1 if $\|x-y\|\neq 1$ for all $x,y\in A$. Let $m_1(\mathbb{R}^n,\|\cdot\|)$ be the supremum of the upper densities of measurable sets avoiding distance $1$, and write $m_1(\mathbb{R}^2)$ for $m_1(\mathbb{R}^2,\|\cdot\|_2)$. Erdős's conjecture.

$$m_1(\mathbb{R}^2)<\frac{1}{4}.$$

The conjecture would improve the known upper bound for the maximal density of a distance-1-avoiding set in the Euclidean plane; at the time of the source, the best known lower bound was $0.2293$ and the best upper bound was approximately $0.25688$.