= Solution
There are two competing extremal-surface topologies. The disconnected candidate $\gamma_1\cup\gamma_2$ has area independent of the separation $L$ and gives $I_c=0$. A connected surface joining the two boundary circles can have smaller area when the gap is small; as $L\to0$, short-distance entanglement across the nearby boundaries makes $I_c$ positive and divergent.
Increasing $L$ makes the connected candidate less favorable. At a critical separation $L=C(R_1,R_2)$ its area equals the disconnected area, and beyond that point the disconnected candidate is minimal. This <entanglement-wedge phase transition> gives
$$
I_c>0\quad(L<C),
\qquad
I_c=0\quad(L>C).
$$
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