Minimum moon mass needed for HIP 41378 f to undergo the full tilting mechanism through a resonance with s₆. Each panel corresponds to a given possible value of s₆ (labels). The minimum moon mass m_min (colour scale) is obtained from Eq. (5) as a function of the unknown parameters J₂ and ωλ of HIP 41378 f. The white curves highlight three mass levels, namely m_min/M = 3 × 10⁻⁴, 6 × 10⁻⁴, and 10 × 10⁻⁴. The grey region is forbidden by the left inequality in Eq. (5). The black dotted line shows the relation between J₂ and ωλ obtained for a homogeneous planet (see Eq. B.1). For reference, the black dots show the parameters (ωλ, J₂) of the Solar System giant planets (Yoder 1995).