Distribution of hooks in self-conjugate partitions

We confirm the speculation that the distribution of t-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length t among the size n self-conjugate partitions is asymptotically normally distributed with mean ${\mu }_t\left(n\right)$ and variance ${\sigma }_t{\left(n\right)}^2$${\mu }_t\left(n\right)\sim \frac{\sqrt{6n}}{\pi }+\frac{3}{{\pi }^2}-\frac{t}{2}+\frac{{\delta }_t}{4}\text{and}{\sigma }_t^2\left(n\right)\sim \frac{({\pi }^2-6)\sqrt{6n}}{{\pi }^3},$ where ${\delta }_t:=1$ if t is odd and is 0 otherwise.