The zero-intersection characterization of asymptotic shadows in the Thurston boundary

Let $X$ be the Teichmüller space of a compact surface equipped with the Teichmüller metric $d$, and let its Thurston compactification be $\overline{X}=X\cup\mathcal{PMF}$. Denote by $S(F)$ the shadow of $F\in\mathcal{PMF}$, and by $i(F,G)$ the geometric intersection number of measured foliations.

Zero-intersection characterization. For any $F\in\mathcal{PMF}$,

$$S(F)=\{G:i(F,G)=0\}.$$

The preceding theorem establishes only the inclusion $S(F)\subset\{G:i(F,G)=0\}$ when $F$ is minimal, while uniquely ergodic foliations have singleton shadows. The asserted equality for every point of the Thurston boundary is therefore a stronger statement whose resolution is not supplied here.