[en] A toroidally linked mirror system has recently been proposed as a neutron source. Two parallel pairs of minimum-B mirror cells are linked by two semitori with cross sections having high ellipticity. Large elongation of the cross-section perpendicular to the toroidal radius of curvature can lead to easier line-bending in the bad curvature region, resulting in relatively lower toroidal beta (β_τ) limits due to short wave-length ballooning instabilities. However, kinetic effects can stabilize such modes with short perpendicular wavelengths, and the most unstable modes are likely to have long wavelengths. The authors explore both regimes to obtain estimates of the critical β_τ limits