Universal enveloping triassociative algebra conjecture for Lie trialgebras

Let $F\colon \mathbf{TriAss} \to \mathbf{TriLie}$ be the functor obtained from the operations in Lemma, sending a triassociative algebra to its associated Lie trialgebra. For a Lie trialgebra $L$, let $U(L)$ denote a proposed universal enveloping triassociative algebra. Universal enveloping triassociative algebra conjecture. The functor $F$ should have a left adjoint $U$ sending $L$ to $U(L)$, and the natural map

$$L\longrightarrow U(L)$$

should be injective. This is the expected analogue of the universal enveloping construction for Lie trialgebras; the source gives no result establishing the adjunction or injectivity.