Erdős et al.'s conjecture on monochromatic copies of nonequilateral triangles

Let a coloring $\chi$ be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains a triangle $T$ if there is a monochromatic copy of $T$, where copies are obtained by translations and rotations in the plane. A triangle is nonequilateral if its three side lengths are not all equal. Erdős et al.'s conjecture. Every coloring $\chi$ contains every nonequilateral triangle $T$. The source states that this conjecture remains open; it concerns whether every two-coloring of the plane necessarily contains a monochromatic copy of each fixed nonequilateral triangle.