Full holonomy conjecture for higher-dimensional Laplacian dynamics

Let $P(t)$ be the state of Laplacian dynamics on $n$ vertices, let $D(P(0))$ denote its degree matrix, and let $X(t)$ solve

$$\dot{X}=-L(P)X.$$

For a base point determined by the dynamics, write $\operatorname{Hol}_p^0$ for the restricted holonomy group.

Full holonomy conjecture. For $d\geq 3$, $n>d$, and initial conditions with $D(P(0))$ not scalar (non-regular graph), the restricted holonomy group of Laplacian dynamics is

$$\operatorname{Hol}_p^0=\mathrm{SO}(d).$$

The preceding finite-time rank criterion proves full holonomy when sufficiently many curvature-generated skew directions are linearly independent. Establishing that this rank condition holds generically for all $d\geq 3$ would require a transversality argument controlling symmetry strata, so the fully generic statement remains conjectural.