Barycentric orbital elements of putative perturber planets. Distributions of barycentric orbital elements of planetary orbits that may result in close encounters (under 5 AU for q_p > 300 AU, under 7.5 AU for q_p > 400 AU, and under 10 AU for q_p > 500 AU and q_p > 600 AU) with five or more present-day extreme trans-Neptunian objects (ETNOs). Each column of panels shows the cumulative results of 2 × 10¹⁰ experiments. From left to right: results of imposing q_p > 300 AU (left column, 20 304 orbits), q_p > 400 AU (5671 orbits), q_p > 500 AU (2635 orbits), and q_p > 600 AU (right column, 56 orbits). Median values are shown as vertical blue lines. Based on the available data on known ETNOs, the presence of massive perturbers well beyond 600 AU is strongly excluded within the context of the hypotheses centered in this study.