Uniqueness conjecture for generalized bridge trisections inducing isotopic trisections

Let $(X,\mathcal K)$ be a pair consisting of a 4-manifold and a knotted surface. A generalized bridge trisection is a generalized bridge decomposition of this pair, and two such trisections may be compared by elementary perturbation and unperturbation moves. Uniqueness conjecture. Any two generalized bridge trisections for $(X,\mathcal K)$ that induce isotopic trisections of $X$ can be made isotopic after a sequence of elementary perturbation and unperturbation moves. This conjecture concerns uniqueness of generalized bridge trisections relative to the induced trisection of the ambient 4-manifold; the source suggests that an affirmative answer may follow from the relevant Cerf theory.