The conjectured exact closed-rich constant of the Fibonacci word

Let $f$ be the infinite Fibonacci word, let $C_f$ be its closed-rich constant, and let $\phi=\frac{1+\sqrt{5}}{2}$ be the golden ratio. Fibonacci closed-rich constant conjecture. For $f$,

$$C_f=\frac{5\phi+3}{\phi^2(\phi^3+2)^2}\approx0.10893.$$

This conjecture is presented as a consequence of the preceding conjectured formula for the first-difference sequence. The paper does not report a proof or a resolution.