Realization conjecture for continua in the interior of torus rotation sets

Let $f\in{\mathrm{Homeo}}_0(\mathbb{T}^2)$ have a rotation set $\rho(F)$ with non-empty interior, and let $C\subseteq{\mathrm{int}}(\rho(F))$ be a continuum. A realization conjecture. There exists a minimal set $M_C$ such that

$$\rho_{M_C}(F)=C.$$

The preceding theorem establishes the claimed genericity on an open and dense subset of the homeomorphisms whose rotation sets have non-empty interior. The authors believe the same realization property holds throughout this class, but the statement is presented as a belief rather than established by the theorem.