Uniform grid-path convergence to radial and chordal SLE8

Let $\mathbb{U}$ be the unit disk and consider a square grid of mesh $\epsilon$. Let the uniform measure be taken on simple grid paths from $0$ to $\partial\mathbb{U}$, and separately on simple paths joining two distinct fixed boundary points. Grid-path SLE8 conjecture. As $\epsilon\searrow 0$, the first measure converges weakly, up to reparameterization, to the trace of radial $\textrm{SLE}_8$, and the second converges weakly, up to reparameterization, to the trace of chordal $\textrm{SLE}_8$. The source notes that the topology on path space was not made precise.