Bryant's small-torsion convergence conjecture for the Laplacian flow

Let $\varphi$ be a closed $G_2$-structure, meaning a positive closed three-form, on a closed $7$-manifold. Consider the Laplacian flow

$$\partial_t\varphi(t)=dd^*\varphi(t),\qquad \varphi(0)=\varphi.$$

A closed $G_2$-structure has sufficiently small torsion when its torsion satisfies the smallness condition intended in the conjecture. Bryant's small-torsion convergence conjecture. If the initial $G_2$-structure is closed and has sufficiently small torsion, then the flow exists for all time and converges to a torsion-free $G_2$-structure. This conjecture is motivated by Joyce's work on torsion-free $G_2$-structures; the source does not state a precise quantitative smallness threshold or a resolution.