I think that inequality will increase but not because of owning capital. Rather because this automation will shift demand away from unskilled workers and into software developers, marketing experts, etc. And I do not think that the supply of skilled worker can adapt fast enough, if any.
The good news (for the US) is that China will be the main casualty. In a way the US has already largely paid this tax to automation by outsourcing factory jobs to China. It is chinese factory workers that are going to be automated the first. Some low skilled jobs that couldn't be outsourced offshore like construction, taxis, etc will suffer but I'd argue a large part of the impact is behind us.
Obviously the tax paid to outsourcing is also having Trump at the white house.