Surfaces in D⁴ with the same boundary and fundamental group

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We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic 4-disk such that they are bounded by the same transverse knot in the standard contact 3-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplectic surface in the standard symplectic 4-disk from a braided surface in a bidisk.