A Question of Norton–Sullivan in the Analytic Case

In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho } h$, where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho }:\mathbb{T}^2\rightarrow \mathbb{T}...