  We take the first step in extending the integrability approach to one-point
functions in AdS/dCFT to higher loop orders. More precisely, we argue that the
formula encoding all tree-level one-point functions of SU(2) operators in the
defect version of N=4 SYM theory, dual to the D5-D3 probe-brane system with
flux, has a natural asymptotic generalization to higher loop orders. The
asymptotic formula correctly encodes the information about the one-loop
correction to the one-point functions of non-protected operators once dressed
by a simple flux-dependent factor, as we demonstrate by an explicit computation
involving a novel object denoted as an amputated matrix product state.
Furthermore, when applied to the BMN vacuum state, the asymptotic formula gives
a result for the one-point function which in a certain double-scaling limit
agrees with that obtained in the dual string theory up to wrapping order.
