Monochromatic parallelograms of prescribed area

Determine whether every finite coloring of the plane contains a monochromatic parallelogram of any prescribed positive area.

Background

The paper constructs a 21-coloring of the plane with no monochromatic parallelogram whose two adjacent side lengths have product 1, which in particular yields a coloring with no monochromatic rectangle of area 1. However, the area of a general parallelogram also depends on the sine of the angle between its adjacent sides. Consequently, excluding parallelograms whose adjacent side-length product equals 1 does not exclude all parallelograms of area 1, and the broader question of whether every finite coloring must contain a monochromatic parallelogram of prescribed area remains unresolved.