Continuous blooming conjecture for convex polyhedral boundaries

Let $S$ be the boundary of a convex polyhedron of dimension $d+1$ in $\mathbb R^{d+1}$. A continuous blooming of $S$ is a nonoverlapping foldout $\overline U\to S$ together with a homotopy $\{\phi_t:\overline U\to\mathbb R^{d+1}\mid 0\leq t\leq1\}$ such that $\phi_0$ is the foldout map, $\phi_1$ is the identity on $\overline U$, each intermediate map is an isometry on the interior and linear on the specified facet components, and the dihedral angles between corresponding facets increase with $t$.

Continuous blooming conjecture. Every convex polyhedral boundary has a continuous blooming.

The conjecture asks whether some nonoverlapping foldout of every convex polyhedral boundary can be continuously folded back without self-intersection while all dihedral angles monotonically increase. As far as the authors know, it is open even for $d=2$.