Kovač and Predojević [KoPr24] have proved that this is true for cyclic quadrilaterals - that is, every set with infinite measure contains four distinct points on a circle such that the quadrilateral determined by these four points has area $1$. They also prove that there exists a set of infinite measure such that every convex polygon with congruent sides and all vertices in the set has area $<1$.
Koizumi [Ko25] has resolved this question, proving that any set with infinite measure must contain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled triangle, all of area $1$.
