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Parallel multiple-input multiple-output (MIMO) broadcast channels have been shown to be separable from an information theoretic point of view, i.e., capacity can be achieved with a strategy that performs separate encoding and decoding on each subchannel subject to a power allocation across subchannels. In this paper, we show that separability does not necessarily hold if the transmit strategy is restricted to linear transceivers. The proof will be done by identifying a rate tuple that is achievable in a certain set of channels with joint treatment of the subchannels, but lies outside the achievable rate region of separate linear precoding. The implications of this result are that any algorithm to optimize transmit strategies in parallel MIMO broadcast channels that is based on both linear transceivers and separate encoding on each subchannel does in fact introduce two suboptimalities aandnd not only one.