The dendriform realization conjecture for refined flow idempotents

Let $\mathscr{F}_t$ be the global series introduced from the parameterized flow series, and let $\mathsf{F}_t=-(1-t)\mathbf{N}_t\mathbf{T}\mathbf{P}_t$ be its dendriform candidate. Refined flow-idempotent realization conjecture. The series $\mathsf{F}_t$ is the dendriform image of $\mathscr{F}_t$. If true, this would extend the corresponding unparameterized construction and provide new Lie idempotents. The source provides no resolution status.