We consider a Gaussian diamond network where a source communicates with the destination through $n$ non-interfering half-duplex relays. We focus on half-duplex schedules that utilize only local channel state information, i.e., each relay has access to its incoming and outgoing channel realizations. We demonstrate that random independent switching, resulting in multiple listen-transmit sub cycles at each relay, while still respecting the overall locally optimal listen-transmit fractions, enables to approximately achieve at least $3/4$ of the capacity of the $2$-relay diamond network. With a single listen-transmit cycle, this fraction drops from $3/4$ to $1/2$. We also provide simulation results that point to the same fractions of capacity being retained over networks with more than $2$ relays.