= No steady displacement layer without ambient supply
Consider entraining plumes supplied with total source <volume flux> $S$, a separate lower-layer ambient inflow $L$, and ceiling exhaust $F=S+L$. Write $P(h)=S+E(h)$, where $E(h)$ is total <fluid entrainment> from the lower layer. <Volume conservation> gives $A\dot h=L-E(h)$. A steady positive lower layer therefore requires $L=E(h)>0$. If $L=0$ and $E(h)>0$ for every $h>0$, no positive steady <displacement-ventilation interface height> exists. In a shifted <pure plume> approximation $P_i(h)=\lambda B_i^{1/3}(h+z_i)^{5/3}$ matched to source flux $S_i$ at $h=0$, this follows immediately from $P_i(h)>S_i$ for $h>0$. This is a statement about the ideal two-layer model, not a prohibition of all possible stratification in a real ventilated room.
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