C2 and C3 are in parallel. They must have the same voltage.
If they are charged as noted, 3V and 5V, as soon as you connect them,
The charges will re-distribute so that both have the same voltage and q2 = c2 * V and q3 = 3 * V
When this happens, the current is limited only by the resistance of the wires.
The schematic shows 3V and 5V, while the text says 2V and 5V ?
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Conceptually:
C2 and C3 are effectively 1 capacitor. he parallel resistor will discharge the parallel combination of C2 and C3 until there is 0 V across them,
and hence 0V across the resistor.
Meanwhile, the voltage on C1 will rise to 10 V, and current will stop flowing.
So, after a long time, there will be no current flowing, C1 will be at 10 V, and C2, C3 and R will have 0V across them.