Open, the site's label (page last edited 16 October 2025). No result settles the question. Three results decide the question for particular triangles and are recorded as partial claims: Shader's refereed theorem of 1976 that every right triangle, and every triangle of two further one-parameter families, has a monochromatic congruent copy in every two-coloring of the plane, on Shader 1976 (accepted on the refereed publication alone); the triangle families of the 1975 colloquium paper of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, on EGMRSS 1975 (claimed, a proceedings volume); and Currier, Moore and Yip's refereed theorem of 2024 that three equally spaced collinear points, and every $(\alpha,2\alpha,x\alpha)$ triangle with $1\le x\le3$, appear monochromatically, on Currier, Moore and Yip 2024 (accepted on the refereed publication alone). No full claim exists, and the standing derives from the claim pages.