The nucleosynthetic isotope dichotomy between carbonaceous (CC) and non-carbonaceous (NC) meteorites has been interpreted as evidence for spatial separation and the coexistence of two distinct planet-forming reservoirs for several million years in the solar protoplanetary disk. The rapid formation of Jupiter’s core within one million years after the formation of calcium-aluminium-rich inclusions (CAIs) has been suggested as a potential mechanism for spatial and temporal separation. In this scenario, Jupiter’s core would open a gap in the disk and trap inward-drifting dust grains in the pressure bump at the outer edge of the gap, separating the inner and outer disk materials from each other. We performed simulations of dust particles in a protoplanetary disk with a gap opened by an early-formed Jupiter core, including dust growth and fragmentation as well as dust transport, using the dust evolution software DustPy. Our numerical experiments indicate that particles trapped in the outer edge of the gap rapidly fragment and are transported through the gap, contaminating the inner disk with outer disk material on a timescale that is inconsistent with the meteoritic record. This suggests that other processes must have initiated or at least contributed to the isotopic separation between the inner and outer Solar System.