Player investment is really tricky to gauge when I deal with Wallflowers. In a game I played a few weeks ago we had this happen.
When a situation came up, and it was Kris' turn to act, he just shrugged and said "I dunno, come back to me." until everyone else had acted, and then spent his action "just watching".
As a player I feel it's part of my responsibility to engage with and involve the other players, so I call out for help from him, as part of my action. That gives him something to do, but after he helps me out on his action, he's back to watching and doing very little else. This is during a storm, and there are a number of different things the GM was pushing out there to deal with, more problems than there are characters. He had choices, he chose not to get involved in the choices.
I try this again with a brief conversation between his character and mine, trying to focus on the little Kris has revealed about the character. Though I think I'm able to entertain him, he's not giving me anything back to work with.
There's a sense that he's shirking his responsibilities as a player to engage in what's going on, and interact with the other players, or do anything to advance his or other people's interests in the game.
That drain can be really hard at times. We've dealt with it by giving him an opportunity to engage, but accepting a "I'm not sure what to do right now" as an action and moving to the next person. Not ideal, I know, but the alternative, where we try to figure out what he can do, and give suggestions until he says of "yeah, I do that", every single time his turn comes around, it can really slow down play and kill the energy of the scene.