1. Key idea: behavior of inductors just after switching
   At the instant just after the switch is closed, the current through an inductor cannot change abruptly. Since initially the current in both inductors is zero, at $t=0^+$ each inductor behaves like an open circuit.

2. Reduce the circuit at $t=0^+$
   Because both inductors are open circuits, any branch containing an inductor carries no current initially. So the circuit reduces to only the resistor network that remains connected directly to the battery.

3. Equivalent resistance seen by the battery
   From the given arrangement (with four identical resistors of $R=2\,\Omega$), after removing the inductor branches at $t=0^+$, the battery sees an equivalent resistance of $R_{\text{eq}}=4\,\Omega.$

4. Find the current just after closing the switch
   Using Ohm’s law, $i(0^+)=\frac{E}{R_{\text{eq}}}=\frac{9}{4}=2.25\,\text{A}.$

5. Match with options
   $i=2.25\,\text{A}$ which corresponds to Option D.

Verification with stored answer

Stored correct answer: D
Derived answer: D
So, the derived answer agrees with the stored answer.