Finite-composition conjecture for Semenov-type shifts

Let $M=\{m_j:j\geq1\}$ be a shift, and set $N_j(M)$ for the associated Semenov counting quantity. A sequence is called decomposable when it satisfies the hypothesis of Definition 1, namely when there are constants and sequences giving the two displayed dyadic approximation conditions in that definition. Finite-composition conjecture. Every shift satisfying the Semenov-type condition

$$\sup N_j(M)<\infty$$

can be written as a finite composition of shifts satisfying the hypothesis of Definition 1. The preceding theorem shows that each decomposable shift induces an isomorphism on $L^p_E$ for every UMD space $E$ and $1<p<\infty$; the supplied text does not state whether the finite-composition assertion is resolved.