= Solution
The conserved total <angular momentum> is $\mathbf J_t=(0,0,2)$, and the <black-hole spin> magnitude is
$$
J_h=\sqrt{\frac34+\frac94}=\sqrt3.
$$
The initial hole is noncollinear with the total vector, so persistent positive alignment <torque> leads to the positive $z$ direction. The final vectors are
$$
\boxed{\mathbf J_{h,f}=(0,0,\sqrt3),\qquad
\mathbf J_{d,f}=(0,0,2-\sqrt3).}
$$
The final disk magnitude is $2-\sqrt3$, smaller than its initial magnitude one, and its direction is parallel to the hole. Their sum remains $(0,0,2)$.
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