Recent studies have shown that only a small fraction of fast-flow bursts observed at mid-tail penetrate into the inner magnetosphere, raising questions regarding their role in particle injections. Motivated by these findings, we compared observations at two radially-aligned Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft in the high-beta nightside plasma sheet to determine which physical parameter controls the penetration efficiency of flow bursts observed at the outer spacecraft. We showed that the inferred plasma tube entropy PV5/3 demonstrates better prediction efficiency than other parameters (e.g., Vx or Bz). Comparing its minimal value at the outer spacecraft during the flow burst to its preflow value at the inner spacecraft allows us to distinguish between penetrating and non-penetrating events. Our results explain the relatively small number of deeply penetrating BBFs and provide a strong argument in favor of the bubble model of fast-flow bursts and plasma injections.