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It is known (Stipsicz-Szab\'o-Wahl) that there are exactly threetriply-infinite and seven singly-infinite families of weighted homogeneousnormal surface singularities admitting a rational homology disk($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor numberzero. Some examples are found by an explicit "quotient construction", whileothers require the "Pinkham method". The fundamental group of the Milnor fibrehas been known for all except the three exceptional families $\mathcalB_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settlethese cases. We present a new explicit construction for the $\mathcal B_2^3(p)$family, showing the fundamental group is non-abelian (as occurred previouslyonly 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 $\mathcalC^3_3(p)$ are abelian, hence easily computed; using the Pinkham method hererequires precise calculations for the fundamental group of the complement of aplane curve.

Rational homology disk smoothings of surface singularities; the exceptional cases