Diagram-independence conjecture for boundary algebras of marked surfaces

Let $(X,M)$ be a marked surface with nonempty boundary, with marked points only on its boundary, and suppose that $(X,M)$ is not a monogon or digon. Assume that there is at least one marked point on each boundary component. A weak Postnikov diagram on $(X,M)$ is a diagram of the type considered in the paper, with boundary algebra associated to it.

Boundary-algebra independence conjecture. The boundary algebra of a weak Postnikov diagram on $(X,M)$ does not depend on the choice of diagram up to isomorphism.

This conjecture extends the established diagram-independence results for disks and annuli, suggesting that the boundary algebra is an invariant of the marked surface rather than of the chosen weak Postnikov diagram. Its status is not resolved in the supplied source context.