Conjecture on Hall-monomial constants in the free Lie algebra of rank 2

Let $L$ be the free Lie algebra of rank $2$, with derivation $\delta$, and let $U^{(1)}$ be the Lie subalgebra specified in the surrounding construction. A Hall monomial is a Hall-basis monomial of $L$, and a constant under $\delta$ is an element annihilated by $\delta$.

Hall-monomial constants conjecture. If $M$ is a Hall monomial that is a constant under the derivation $\delta$, then

$$M\in U^{(1)}.$$

The conjecture is motivated by computations showing that constants of degree at most $7$ belong to the relevant Lie subalgebra, but no resolution is supplied in the source.