This paper investigates the simplification problem in Gaussian Half-Duplex (HD) diamond networks. The goal is to answer the following question: what is the minimum (worst-case) fraction of the total HD capacity that one can always achieve by smartly selecting a subset of $k$ relays, out of the $N$ possible ones? We make progress on this problem for $k=1$ and $k=2$ and show that for $N=k+1, \ k \in \{1,2\}$ at least $\frac{k}{k+1}$ of the total HD capacity is always {approximately (i.e., up to a constant gap)} achieved. Interestingly, and differently from the Full-Duplex (FD) case, the ratio in HD depends on $N$, and decreases as $N$ increases. For all values of $N$ and $k$ for which we derive worst case fractions, we also show these to be {approximately} tight. This is accomplished by presenting $N$-relay Gaussian HD diamond networks for which the best $k$-relay subnetwork has {an approximate} HD capacity equal to the worst-case fraction of the total {approximate} HD capacity. Moreover, we provide additional comparisons between the performance of this simplification problem for HD and FD networks, which highlight their different natures.