Protons (ionized hydrogen) in the solar wind frequently exhibit distinct temperatures ($T_{perp p}$ and $T_{parallel p}$) perpendicular and parallel to the plasmas background magnetic-field. Numerous prior studies of the interplanetary solar-wind have shown that, as plasma beta ($beta_{parallel p}$) increases, a narrower range of temperature-anisotropy ($R_pequiv T_{perp p},/,T_{parallel p}$) values is observed. Conventionally, this effect has been ascribed to the actions of kinetic microinstabilities. This study is the first to use data from the Magnetospheric Multiscale Mission (MMS) to explore such $beta_{parallel p}$-dependent limits on $R_p$ in Earths magnetosheath. The distribution of these data across the $(beta_{parallel p},R_p)$-plane reveals limits on both $R_p>1$ and $R_p<1$. Linear Vlasov theory is used to compute contours of constant growth-rate for the ion-cyclotron, mirror, parallel-firehose, and oblique-firehose instabilities. These instability thresholds closely align with the contours of the data distribution, which suggests a strong association of instabilities with extremes of ion temperature anisotropy in the magnetosheath. The potential for instabilities to regulate temperature anisotropy is discussed.
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