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I was curious and looked it up, and well, seems like you are correct, but also it's all a bit of a mess and the MGB is part of a trio of companies of which one is the infrastructure (owned entirely by government), one is the management (owned half by the government, half by BVZ Holding), and one is MGB itself (presumably the actual trains etc.? Not sure) which is mostly owned by BVZ Holding.
So... yeah, we got some complicated company structures there too.
I think MGB might be fully privately owned but I'm not sure.