Bad-tree generation conjecture for graded identities of upper triangular Lie algebras

Let $G$ be a group and let $(UT_n^{(-)},\eta)$ be an elementary $G$-grading, where $UT_n^{(-)}$ denotes the Lie algebra of upper triangular $n\times n$ matrices. A tree $\mu$ is a parenthesized sequence of elements of $G$, with length $\ell(\mu)$ equal to the number of its leaves. For each tree $\mu$, let $f_\mu$ be the associated multilinear Lie monomial, and call $\mu$ an $\eta$-bad tree if it is not $\eta$-good. Bad-tree generation conjecture. The graded polynomial identities of $(UT_n^{(-)},\eta)$ follow from the polynomials $f_\mu$ for all $\eta$-bad trees of length at most $n$. This conjecture asserts that all graded identities are generated by these bad-tree identities; the supplied text gives no resolution or further general result.