Existence of a lifted Markov trace for cubic Hecke algebras

Let $Q(x)=x^3-\alpha x^2-\beta x-1$, and let $H(Q,n)=\mathbb{C}[B_n]/(Q(b_j);\,j=1,\dots,n-1)$ be the cubic Hecke algebra. The invariant $I_{(\alpha,\beta)}$ is defined by a Markov trace on these algebras. Lifted Markov trace conjecture. There exists a Markov trace on $H(Q,n)$ taking values in an algebraic extension of $\mathbb{Z}[\alpha,\beta]$ that lifts the Markov trace underlying $I_{(\alpha,\beta)}$. This would provide a genuine two-parameter extension of the invariant constructed in the paper; the source does not state whether the conjecture has been resolved.