Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer $k$, there exists an $\text{MSO}_1$-transduction that given a graph of linear cliquewidth at most $k$ outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of $k$. A direct corollary of this result is the equivalence of the notions of $\text{CMSO}_1$-definability and recognizability on graphs of bounded linear cliquewidth.