This vaccine relies on inducing immunity to an HIV peptide* (i.e. a ~9 amino acid cleavage product of an HIV protein) that is predicted to bind to HLA molecules. At first blush, this sounds like a great idea--let's immunize people to the very thing that the immune system is supposed to recognize as bad! Unfortunately, this has been tried lots of times and usually doesn't work very well, if at all. It's been tried in cancer, it's been tried in HIV, it's been tried in practically any system you can imagine. I'm not saying that peptide vaccines don't ever work (hell, my lab has worked on one)--I'm just saying that they have had lots of promise and limited successes. In fact, animals and patients have been immunized to many cleavage products of many cancer associated antigens to little, if any, effect—or effects in animals but not in humans. We can hem and haw all day about what the specific studies show, but suffice it to say that peptide vaccines are an old idea. Perhaps Immunity will get lucky here, but I'd be surprised if there isn't a lab somewhere that has tried immunizing animals to every single breakdown product of HIV proteins already.
Immunity's white paper says that their putatively immunogenic human peptide cannot be tested in animal models because it will only bind to human HLA. So they are planning to use a mouse-analog of the human peptide to do their animal testing (or have already done this?). That is, they are planning to use (or maybe have already used?) a completely different peptide (probably 1-3 amino acids different from the human version) in animal studies than what they would use in patients. This may be a significant difference... or it might not be. There's no way to tell a priori. A more classical and IMO better approach would be to identify T cell receptor clones from patients with HIV (or without HIV) that bind to these peptides (using tetramer staining, phage display or some other high-throughput technology), demonstrate that humans are capable of creating T cells that recognize their putatively immunogenic peptide followed by T cell killing assays on cells presenting their peptide of interest. There are other approaches, but that is a commonly used one in the field.
* in some places the white paper says "peptides" and in other it says "peptide" so it's not clear if we are talking about 1 peptide or multiple. I think they are mostly talking about multiple peptides, because they want to immunize people no matter their HLA type. Different HLA types bind to different peptides. So if you have HLA A02 you will bind different peptides than if you have HLA A24... and so on for all the different HLA types.