I don't know what they mean by conscious experience, however while writing this reply, there is awareness of what is happening at this moment, it is something involving thoughts, sensations of my body and my hands pressing buttons additionally there is awareness of visual space too, and making sense of this along with characters appearing on the screen. Now making sense of what's going on, does have a social angle to it.
However the real fact of being aware of all this is clearly at a much fundamental level. To understand we can imagine another thought experiment. By imagining a child when he is just born, Now assuming there are no concepts at that moment, the child is very aware of the experience of what's happening, that to me is consciousness which clearly precedes all concepts that will come afterwards, including the concept of the "self" and living and experiencing life on behalf of a person in whatever social context that person happen to find himself.