Cosmic string gravitational wave backgrounds at LISA: II. Reconstruction of conventional signals over astrophysical foregrounds

We study the reconstruction of conventional cosmic-string signals with LISA in the presence of all major known astrophysical foregrounds expected in the LISA band. These include stellar-origin black-hole binaries (SOBHBs), galactic (WDs) and extragalactic (ExWDs) white dwarfs, extreme-mass-ratio-inspirals (EMRIs), and massive black-hole binaries (MBHBs). Using the Simulation-based Inference package GWBackFinder, we perform a joint inference on the LISA noise, foregrounds, and signal, across a range of injected string tensions $G\mu$. We find that reconstructing tensions with an error $\lesssim 10\%$ requires values as large as $G\mu \gtrsim 10^{-11}$, i.e. a factor $\sim10^5$ larger than previous estimates with no foregrounds, and $\sim 10^2$ larger compared to estimates accounting only for SOBHB and WD foregrounds. This work is the second in a series initiated in Ref. arXiv:2508.05395, which aims to quantify LISA's ability to measure representative cosmic-string models.