Answer: No
Step-by-step explanation:
I pretty sure the current shouldn't flow because even though the light bulbs are intact, because at least one of the switches have to close for it to connect with the other part of the circuit.
The current can loop around and light up the light bulb, meaning the current has flowed all the way in a loop through that light bulb and not abruptly stopped in between since the switch hadn't closed.
There are 3 batteries shown in the image, so it would be enough for the current to flow through all 3 light bulbs, if only all 3 switches were closed.
Hope that helped a lot.
:)