Conjecture that constants equal the algebra U^(1) in the free Lie algebra of rank 2

Let $L$ be the free Lie algebra of rank $2$, let $\delta$ be the derivation under consideration, and let $L^\delta$ denote the algebra of constants of $\delta$:

$$L^\delta=\{u\in L\mid \delta(u)=0\}.$$

Let $U^{(1)}$ be the algebra constructed in the preceding discussion.

Constants-equality conjecture. The algebra of constants is exactly $U^{(1)}$:

$$L^\delta=U^{(1)}.$$

The claim is presented as a stronger conjecture supported by the results of the section; the supplied text gives no evidence that it has been resolved.