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 ${\mu }_t(n)\sim \frac{\sqrt{6n}}{\pi }+\frac{3}{{\pi }^2}-\frac{t}{2}+\frac{{\delta }_t}{4}$ and variance ${\sigma }_t^2(n)\sim \frac{({\pi }^2-6)\sqrt{6n}}{{\pi }^3},$ where ${\delta }_t:=1$ if $t$ is odd, and is 0 otherwise.