Eh, I think "no" to the question of emitter current isn't quite "no" to the question of the base-collector diode I think. The latter sort of conduction may or may not be considered to be inside what john1 describes as working voltages (I'm not quite awake ATM sorry), it's not quite standard usage. It crops up IIRC when transistors go into saturation, which is at least ordinary when the emitters are conducting. Whatever. If an NPN's base is ~0.6v more positive than its collector, it conducts that way, and if a PNP's collector is ~0.6v more positive than its base, it conducts that way. Something along those lines.
Also there's the question of what sort of voltage/current those top and left terminals have. Because the diagram looks as though the top terminal would be +ve, in which case you're showing PNP transistors where you'd surely want NPN transistors- unless I'm missing something real unusual, neither transistor will pass current with grounded emitter but +ve base and collector, they'd be reverse-biased or whatever. Or blown up. In fact they might be blown up anyway as there's no resistors in sight and we haven't been told the terminals go to current sources.
Thos various things aside though, no I'm not aware of any way that the disconnected emitter could pass current.