One-parameter subgroup conjecture for the heat equation symmetry pseudogroup

Let $G$ be the point-symmetry pseudogroup under consideration, with splitting $G=G^{\rm ess}\ltimes G^{\rm lin}$ and Lie algebras $\mathfrak g$ and $\mathfrak g^{\rm ess}$. For $\Phi\in G$, write

$$\Phi=\Phi^{\rm ess}\circ\Phi^{\rm lin}$$

for its decomposition into $G^{\rm ess}$- and $G^{\rm lin}$-components. One-parameter subgroup conjecture. The transformation $\Phi$ belongs to a one-parameter subgroup of $G$ if and only if $\Phi^{\rm ess}$ belongs to a one-parameter subgroup of $G^{\rm ess}$; equivalently,

$$\Phi\in\exp(\mathfrak g)\quad\Longleftrightarrow\quad\Phi^{\rm ess}\in\exp(\mathfrak g^{\rm ess}).$$

If true, this would justify calling the reflection $\mathscr J=(t,x,u)\mapsto(t,-x,u)$ a pseudo-discrete point symmetry of the heat equation, because its essential component is not contained in a one-parameter subgroup of $G^{\rm ess}$. The source gives no resolution of the conjecture.