Low-degree convergence of relaxed minimal-deformation surface elements

Let $\Gamma^0 \subset \mathbb{R}^3$ be a smooth closed oriented surface and $u : \mathbb{R}^3 \times [0, T] \to \mathbb{R}^3$ a smooth velocity field. Assume that the relaxed minimal-deformation equations $$\partial_t X = v, \qquad v \cdot (n \circ X) = (u \circ X) \cdot (n \circ X), \qquad \partial_t X - \Delta_{\Gamma^0} X = \kappa \, (n \circ X) \qquad \text{on } \Gamma^0,$$ with $X(\cdot, 0) = \operatorname{id}$, have a smooth solution $(X, v, \kappa)$, where $\Gamma(t) = X(\Gamma^0, t)$ has unit normal $n$, the scalar $\kappa$ is a Lagrange multiplier, and $X(\cdot, t)$ and its inverse are smooth for all $t \in [0, T]$.

Fix $k \in \{1, 2, 3\}$. Let $\Gamma_h^0$ be the degree-$k$ nodal interpolants of $\Gamma^0$ on an admissible, shape-regular, quasi-uniform family of curved triangulations. For each curved triangle $K$ let $K_f$ be the flat triangle with the same vertices and $F_K : K_f \to K$ its polynomial parametrization, and assume $$\max_K \bigl( \|F_K\|_{W^{k,\infty}(K_f)} + \|F_K^{-1}\|_{W^{1,\infty}(K)} \bigr) \le C_0$$ with $C_0$ independent of $h$. Let $S_h$ be the continuous parametric finite element space of degree $k$ on $\Gamma_h^0$. The semidiscrete scheme seeks $(X_h, v_h, \kappa_h) \in S_h^3 \times S_h^3 \times S_h$ with $X_h(\cdot, 0) = \operatorname{id}$, $\partial_t X_h = v_h$ and $$\int_{\Gamma_h^0} \bigl(v_h - u(X_h, t)\bigr) \cdot N_h \, \chi = 0 \quad (\chi \in S_h), \qquad \int_{\Gamma_h^0} v_h \cdot \eta + \int_{\Gamma_h^0} \nabla_{\Gamma_h^0} X_h : \nabla_{\Gamma_h^0} \eta = \int_{\Gamma_h^0} \kappa_h N_h \cdot \eta \quad (\eta \in S_h^3),$$ where $N_h = \tilde n_h \circ X_h$ and $\tilde n_h$ is the elementwise normal of the surface $X_h(\Gamma_h^0)$, which may jump across edges. All integrals are exact; there is no normal averaging or other stabilization. Errors are measured through the lift $X_h^\ell = X_h \circ a_h^{-1}$, where $a_h : \Gamma_h^0 \to \Gamma^0$ is the closest-point projection.

Gao, Li and Tang proved that for $k \ge 4$ the scheme has a unique solution on $[0, T]$ for small $h$, with $\sup_{t \le T} \|X_h^\ell - X\|_{L^2(\Gamma^0)}^2$ plus an $L^2(0,T;H^1)$ term for the tangential error bounded by $C h^{2k}$, and observed convergence numerically for $k = 1, 2, 3$.

Problem. For each $k = 1, 2, 3$, does the scheme, for every such smooth solution and mesh family, admit a unique solution with nondegenerate evolving elements on $[0, T]$ for all sufficiently small $h$, and satisfy $$\sup_{0 \le t \le T} \|X_h^\ell(\cdot, t) - X(\cdot, t)\|_{L^2(\Gamma^0)} \to 0 \qquad (h \to 0)?$$ A positive answer should come with a quantitative error bound; a negative one is a counterexample under these assumptions.