Generalized SLE-bubble welding conjecture with a boundary insertion

Fix $W_1\geq\frac{\gamma^2}{2}$ and $W>2$. Let $\mathrm{Bubble}_{\mathbb H}(p)$ denote the space of bubbles in $\mathbb H$ rooted at $p$, and let $\mathrm{LF}_{\mathbb H}^{(\beta,p)}$ be the Liouville field measure with a boundary insertion of weight $\beta$ at $p$. Write $\mathrm{QD}_{0,1}(\gamma,\beta_W;\ell)$ and $\mathrm{QD}_{1,1}(\gamma,\beta_W;\ell)$ for the indicated quantum disks with boundary length $\ell$, and let $\mathbf m_{\mathbb H}$ and $\mathbf m_{\mathbb H,0}$ be the stated Haar measures. Generalized SLE-bubble welding conjecture. There exist a $\sigma$-finite infinite measure $\mathrm{SLE}^{\mathrm{bubble}}_{\kappa,p}(W,W_1)$ on $\mathrm{Bubble}_{\mathbb H}(p)$ and constants $C\in(0,\infty)$ such that

$$\mathbf m_{\mathbb H}\ltimes\left(\int_0^\infty\mathcal M^{\mathrm{disk}}_{0,2}(W_1;\cdot,\ell)\times\mathrm{QD}_{0,1}(\gamma,\beta_W;\ell)d\ell\right)=C\cdot\mathrm{LF}^{(\beta_{2W_1+W},p)}_{\mathbb H}\times\mathrm{SLE}_{\kappa,p}^{\mathrm{bubble}}(W,W_1)\,dp,$$

and, for a suitable constant $C\in(0,\infty)$,

$$\mathbf m_{\mathbb H,0}\ltimes\left(\int_0^\infty\mathcal M^{\mathrm{disk}}_{0,2}(W_1;\cdot,\ell)\times\mathrm{QD}_{0,1}(\gamma,\beta_W;\ell)d\ell\right)=C\cdot\mathrm{LF}_{\mathbb H}^{(\beta_{2W_1+W},0)}(d\phi)\times\mathrm{SLE}_{\kappa,0}^{\mathrm{bubble}}(W,W_1).$$

Here $\mathbf m_{\mathbb H,0}$ is Haar measure on the conformal automorphism group of $\mathbb H$ fixing $0$. This would generalize the paper's welding theorems to a quantum disk with one general boundary insertion. The associated bubble law is poorly understood, and the conjecture is open.