Chordal SLE${}_6$ explorations of a quantum disk

We consider a particular type of $\sqrt{8\mathrm{/}3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE${}_6$ curve $\eta$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta ([0,t])$ for each time $t\ge 0$ as well as the law of the left/right $\sqrt{8\mathrm{/}3}$-quantum boundary length process for $\eta$.