Recently Vernstrom et al. (2021, MNRAS) claimed the first definitive detection of the synchrotron cosmic web, obtained by ‘stacking’ hundreds of thousands of pairs of close-proximity clusters in low-frequency radio observations and looking for a residual excess signal spanning the intracluster bridge. A reproduction study by Hodgson et al. (2022, PASA, 39, e013), using both the original radio data as well as new observations with the Murchison Widefield Array, failed to confirm these findings. Whilst the detection remains unsure, we here turn to stacking a simulated radio sky to understand what kind of excess radio signal is predicted by our current best cosmological models of the synchrotron cosmic web. We use the FIlaments & GAlactic RadiO (FIGARO; Hodgson et al. 2021a, PASA, 38, e047) simulation, which models both the synchrotron cosmic web as well as various subtypes of active galactic nucleii and star-forming galaxies. Being a simulation, we have perfect knowledge of the location of clusters and galaxy groups which we use in our own stacking experiment. Whilst we do find an excess radio signature in our stacks that is attributable to the synchrotron cosmic web, its distribution is very different to that found by Vernstrom et al. (2021, MNRAS). Instead, we observe the appearance of excess emission on the immediate interiors of cluster pairs as a result of asymmetric, ‘radio relic’-like shocks surrounding cluster cores, whilst the excess emission spanning the intracluster region—attributable to filaments proper—is two orders of magnitude lower and undetectable in our experiment even under ideal conditions.