Eisenstein isoperimetric inequality for Fibonacci colorings

Let $\beta=a+b\alpha$ and $\beta'=a'+b'\alpha$ be primitive Eisenstein integers, and write $\beta\preceq\beta'$ when $b\leq b'$. Let $\beta$ be a Fibonacci Eisenstein integer, let $\beta'\in(0,1)$ be a different Eisenstein integer with $\beta\preceq\beta'$, and let ${\cal T}(\beta')$ be the associated triangulation. Let ${\cal C}={\cal C}(\beta)$ be the continued-fraction coloring associated with $\beta$, and let ${\cal C}'$ be an arbitrary good coloring of ${\cal T}(\beta')$. For a coloring ${\cal D}$, let $\eta({\cal D})=f^2/F$. Eisenstein isoperimetric inequality for Fibonacci colorings. One should have

$$\eta({\cal C})<\eta({\cal C}').$$

The claim is motivated by computations for the first ten examples and by examples in which the continued-fraction coloring minimizes the fold count. It would establish the Eisenstein isoperimetric inequality within the continued-fraction family, but the source reports no proof.