MSO obstruction characterization for bounded linear cliquewidth

Let $\mathscr{C}$ be a class of graphs. A class is bounded linear cliquewidth when its linear cliquewidth is bounded, and $\mathsf{CMSO}$ denotes counting monadic second-order logic.

Linear cliquewidth CMSO obstruction conjecture. A class of graphs $\mathscr{C}$ has bounded linear cliquewidth if and only if the class of trees cannot be $\mathsf{CMSO}$-transduced from $\mathscr{C}$.

The analogous characterization for cliquewidth is known with the class of all graphs in place of trees, but the linear-cliquewidth version remains open according to the survey.