The capacitor network figure is not visible in the prompt, so the exact topology of the circuit is missing. Without the actual arrangement of the capacitors between $A$ and $B$, the value of $C$ cannot be derived uniquely from first principles.

However, since this is an MCQ with given options and a stored correct answer, I can only compare against that stored answer.

1. The required equivalent capacitance is given as $C_{eq}=1\,\mu F.$

2. The unknown capacitor is one of the following: $\frac{31}{23}\,\mu F,\quad \frac{32}{23}\,\mu F,\quad \frac{33}{23}\,\mu F,\quad \frac{34}{23}\,\mu F.$

3. The stored correct answer is B, i.e. $C=\frac{32}{23}\,\mu F.$

Because the network diagram is absent, I cannot independently verify this by circuit reduction.

Hence, based only on the provided stored answer: $\boxed{C=\frac{32}{23}\,\mu F}$