Uniqueness conjecture for generalized bridge trisections with fixed underlying trisection

Let $(X,\mathcal K)$ be a pair consisting of a 4-manifold and a knotted surface, and fix a trisection of $X$. A generalized bridge trisection of $(X,\mathcal K)$ is compared with another one by perturbation and unperturbation moves, where a perturbation is a finite sequence of elementary perturbations. Uniqueness conjecture. Any two generalized bridge trisections for $(X,\mathcal K)$ with the same underlying trisection for $X$ become isotopic after a finite sequence of perturbations and unperturbations. This extends the known uniqueness result for bridge trisections of knotted surfaces in $S^4$ to generalized bridge trisections in the setting of a fixed underlying trisection of $X$.