Main results for NGC 1261. Left: distribution of the modelled properties of the stars belonging to NGC 1261 and its stream as a function of ϕ₁ after enforcing a 60% lower limit on the membership probability. The error bars represent the measurement uncertainties. The black dashed line represents the best-fit track used to model the mean value for the respective property at a given ϕ₁. The results from the N-body simulation in each of the four dimensions are in blue. Stars with v_𝑔sr values inconsistent with the N-body simulation are outlined in red. Top middle: spatial distribution of the same stars plotted in the left panel. The dashed circles show the outlines of the fields targeted by S⁵. The solid circle shows the extent of r_J. Stars plotted using the square markers are those that show different CMD properties from the cluster, as shown in Fig. 4. Top right: 2D density plot of the distribution of potential member stars around NGC 1261. Stars within r_J were removed when creating this plot to enhance the density contrast. Bottom right: 2D histogram showing the density of simulation particles on the sky. The vertical red lines indicate the same window on the sky as shown in the top panels.