I think it more likely that, for most people who rely on State media, the penny won't drop until the current State does. As long as Putin controls what people know, the majority will back him -- or, at least, enough will that any Opposition is ineffective. The blocking of the BBC, as well as Twitter, Facebook etc, illustrates this.
There's a limit on the extent to which the flow of information can be controlled, for sure. If, for example, the War drags on, and continues to cost Russians dear in terms of lives lost, then no amount of censorship can cover up from the families of the dead soldiers that they're dead (or at the very least permanently out of contact).