Compatibility conjecture for the Grossman–Larson coproduct and rooted-tree coproduct

Let $\mathcal H$ be the Hopf algebra of rooted trees, let $H_{GL}$ be the Grossman–Larson Hopf algebra, and let $\Delta$ and $S_{GL}$ denote the relevant coproduct and antipode, with $\Delta_{GL}$ the coproduct transported to $H_{GL}$. For a linear map $m^{2,4}$ defined by

$$m^{2,4}(a\otimes b\otimes c\otimes d)=a\otimes c\otimes bd,$$

Compatibility conjecture. For any rooted tree $t$,

$$(\Delta_{GL}\otimes\operatorname{id}_{\mathcal H})\circ\Delta(t)=m^{2,4}\circ(\Delta\otimes\Delta)\circ\Delta_{GL}(t)$$

and

$$\Delta\circ S_{GL}(t)=(S_{GL}\otimes\operatorname{id})\circ\Delta(t).$$

These identities would give an analogue of the corresponding compatibility between the Kreimer and Grossman–Larson structures. The source presents them as a question, so their resolution is not established here.