The union-of-Reuleaux-frames diameter conjecture

Let $F_1$ and $F_2$ be frames of Reuleaux polygons in the Euclidean plane, lying nonidentically, and suppose that

$$\operatorname{diam}(F_1)=\operatorname{diam}(F_2)=d.$$

Union-of-Reuleaux-frames diameter conjecture. Then

$$\operatorname{diam}(F_1\cup F_2)>d.$$

The paper identifies this as an unproved conjecture used in its planar diameter argument; no resolution is supplied in the given text.