The laminarity conjecture for higher bifurcation currents

Let $\Lambda$ be a parameter space of dimension $D$, let $T_{\mathrm{bif}}$ be the bifurcation current, and let $U\subset\Lambda$ be an open set where $D-k$ critical points, counted with multiplicity, lie in attracting basins. Laminarity conjecture. The current $T_{\mathrm{bif}}^k$ is laminar in $U$ and locally uniformly laminar outside a closed analytic set. The claim predicts geometric laminar structure for higher bifurcation currents under attracting-basin hypotheses; the source does not state that it has been proved.