Absence of nontrivial symmetries to the heat equation in Goursat groups of dimension at least $4$

Using the extension method, we study the one-parameter symmetry groups of the heat equation $\partial_{t} p=\Delta p$, where $\Delta=X_{1}^{2}+X_{2}^{2}$ is the sub-Laplacian constructed by a Goursat distribution $\operatorname{span} (\lbrace X_{1},X_{2} \rbrace)$ in $\mathbb{R}^n$, where the vector fields $X_1$ and $X_2$ satisfy the commutation relations $[X_{1},X_{j}]=X_{j+1}$ (where $X_{n+1}=0$) and $[X_{j},X_{k}]=0$ for $j \geq 1$ and $k \geq 1$. We show that there are no such groups for $n \geq 4$ (with exception of the linear transformations of solutions which are admitted by every linear equation).