If you take what the As have said, primarily to LE & FBI, but also in news reports, and you put it all together, it appears the entire family has always functioned in this dynamic where lying and hiding things from each other is a basic and important part of their foundation. There's some evidence in the email exchanges that this extends into Cindy's family (brother, mother, aunt).
They all seem to have no sense that there is anything at all wrong with lying, instead viewing it as not only okay, but a very important part of the way of getting through life.
Along with this we've seen both a lack of respect for other people, an overstepping of those boundaries, and a lack of regard for the law. (Lee and the dog lady, George ordering people off public property and throwing items that belong to others without regard for possible damage, George and Cindy lying to LE and FBI)
Lee is young enough to change with long-term therapy. George and Cindy are old enough to have little to no chance of changing even with long-term therapy.
Telling the truth for them would require really strong motivation. Sadly, none of them have displayed any signs at all that justice for the victim, their grandchild/niece, or generally 'doing the right thing', is a motivator for them, and that is the motivator for telling the truth and being open in any criminal investigation. (This is the primary motivator for the people who have been interviewed by LE in this case if you think about it.)
George and Lee fear Cindy, and they do what she tells them to do. Cindy's motivator is preservation in her mind of her self-image as a good mother. She's completely desperate in this pursuit. It consumes her. She goes to dramatic extremes with it (Casey's very obvious pregnancy at brother's wedding and Cindy saying not only was Casey not pregnant, she hadn't even had sex!!) The last we heard from Cindy was on LKL the night before Caylee's remains were found. We watched Cindy over months straying farther and farther from the truth, culminating in what we saw on that LKL show. It was shocking, and she showed no embarrassment, no hesitation - she was deeply immersed in her extensive web of lies and self-deception. We saw her defiantly, aggressively defending it.
Fear of being charged, arrested, convicted, going to jail, haven't worked all along as motivators for the As. I can't think of a reason why any of those would suddenly start working as motivators now, and immunity doesn't take them away completely anyway.
Put it all together, and no, immunity won't give these people what they need to tell the truth and to be open.
MOO
Sorry for being long-winded. I was thinking out loud. The topic of *can* Cindy, and therefore George and Lee, be open, and tell the truth, is of great interest to me. I want Caylee's story to be told. I want to know what happened to her. It is really all that's left we can do for Caylee, and she deserves that. Justice will come from Caylee's story being told. Casey will never tell Caylee's story - only George and Cindy and Lee can. I want them to, I just don't think they will.