Homotopic exact controllability conjecture for the controlled wave maps equation

Let $(\mathcal{M},g)$ be a Riemannian manifold. The controlled wave maps equation is a map $\phi:\mathbb{R}\times\mathbb{S}^1\to\mathcal{M}$, with states in $H^1_x\times L^2(\mathbb{S}^1)$ and an $\mathbb{R}^k$-valued control $f\in L^2_{t,x}([0,T]\times\mathbb{S}^1)$. Two states $u[0]$ and $v[0]$ are considered homotopic when their spatial components $u(x)$ and $v(x)$ are homotopic as maps from $\mathbb{S}^1$ to $\mathcal{M}$. Homotopic exact controllability conjecture. For every pair of homotopic states $u[0]$ and $v[0]$ in $H^1_x\times L^2(\mathbb{S}^1)$, there exist $T>0$ and a control $f\in L^2_{t,x}([0,T]\times\mathbb{S}^1)$ such that the unique solution of the controlled wave maps equation with initial state $\phi[0]=u[0]$ satisfies $\phi[T]=v[0]$. This conjectures that the only global obstruction to exact controllability is the homotopy class of the spatial loop; it extends the established controllability result for the spherical target and is motivated by the necessity of preserving homotopy under continuous evolution. The conjecture remains open in the general Riemannian-target setting.