The lonely pseudo-Anosov conjecture

For a finite type surface $S$, let $f\colon S\to S$ be a pseudo-Anosov homeomorphism, and let $(X,q)$ be the associated complex structure and holomorphic quadratic differential. Call $f$ lonely when $\langle f\rangle$ has finite index in ${\rm Aff}_+(S,q)$. Lonely pseudo-Anosov conjecture. There exist lonely pseudo-Anosov homeomorphisms. In fact, lonely pseudo-Anosov homeomorphisms are generic. This claim motivates the study of Veech groups associated to pseudo-Anosov monodromies; the source provides no resolution status for it.