No. Let's assume it's not fake and look for contradictions. Here's how it would be explained if this spacewalk actually happened in space.
The astronaut in view is climbing along something, and climbing is a bit of a choppy motion. When he moves forward then comes to a stop at each handhold, the strap moves "upward" in the field of view. That helmet camera is pointing straight forward and, in reference to the background, the wearer is also climbing up. So the strap is not exactly moving up as if by buoyancy, but rather it is moving forward because of its inertia. When it stretches as far as it can go it stops, then retracts as the astronaut continues moving forward.
Also, that strap tends to be sticking out more than "up" when the astronaut is stationary, indicating that part of the reason it's not staying "when the astronaut pushes it out of the way" is that it's a somewhat rigid strap.
There's not any contradictions in your buoyancy idea as far as I can tell but also not any contradictions in "what NASA wants you to believe," so this video clip can't really be used to prove or disprove the spacewalk.