  It is known (Stipsicz-Szab\'o-Wahl) that there are exactly three
triply-infinite and seven singly-infinite families of weighted homogeneous
normal surface singularities admitting a rational homology disk
($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor number
zero. Some examples are found by an explicit "quotient construction", while
others require the "Pinkham method". The fundamental group of the Milnor fibre
has been known for all except the three exceptional families $\mathcal
B_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settle
these cases. We present a new explicit construction for the $\mathcal B_2^3(p)$
family, showing the fundamental group is non-abelian (as occurred previously
only for the $\mathcal A^4(p), \mathcal B^4(p)$ and $\mathcal C^4(p)$ cases).
We show that the fundamental groups for $ \mathcal C^3_2(p)$ and $\mathcal
C^3_3(p)$ are abelian, hence easily computed; using the Pinkham method here
requires precise calculations for the fundamental group of the complement of a
plane curve.
