Lie point symmetry conjecture for the Ricci flow

Let $(M^n,g)$ be a Riemannian manifold with $n\geq 2$, and write $g=(g_{ij})$ in local coordinates $(x^1,\ldots,x^n)$. Let $t$ denote time, and let $\xi^1,\ldots,\xi^n$ be arbitrary smooth functions of $x^1,\ldots,x^n$. The Ricci flow is the evolution equation for the metric $g$ given by $\partial_t g_{ij}=-2\operatorname{Ric}_{ij}$. Lie point symmetry conjecture. The Lie algebra of classical, or Lie point, symmetries of the Ricci flow is spanned by

$$X_1=\frac{\partial}{\partial t},\qquad X_2=t\frac{\partial}{\partial t}+\sum_{i=1}^n\sum_{j\geq i}^n g_{ij}\frac{\partial}{\partial g_{ij}},$$

and

$$X_{k+2}=\xi^k\frac{\partial}{\partial x^k}-\sum_{i=1}^n\sum_{j\geq i}^n\left(g_{ki}\frac{\partial\xi^k}{\partial x^j}+g_{kj}\frac{\partial\xi^k}{\partial x^i}\right)\frac{\partial}{\partial g_{ij}},$$

where $k\in\{1,\ldots,n\}$. The Ricci flow admits at least this Lie algebra of symmetries, and the conjecture asserts that it admits no additional Lie symmetries. The time translation, scaling, and coordinate covariance symmetries arise from direct calculations and the covariance of the Ricci flow; what remains is to establish that the displayed algebra is also sufficient and hence complete.