Howie from the Immunity Project here. Thanks for your question! We're really just focusing on the preventive aspect of this vaccine as we head into our Phase I human clinical trials.
Since the Immunity Project concept relies on training the immune system to mount a successful response against HIV, why would it not present some benefit to an already-infected person whose infection is otherwise controlled with drugs?
You may just be focusing on the preventative aspect, but there are many millions worldwide who are concerned with controlling HIV without depending on expensive antiviral drugs that in many cases are completely unavailable.
One of the reasons it would be very hard to justify testing this first as a cure is because AIDS patients are by definition immunodeficient. Their Helper T cells (the CD4s mentioned) are being rapidly co-opted by the virus and subject to destruction. In such an environment it is difficult to mount an immune response because the Helper T cells are so instrumental to enabling the cytotoxic immune responses instigated by this study (the CD8 cells are your cytotoxic "killer" T cells).
Any trial run on humans already suffering from AIDS would be muddied by this effect, where the already compromised immune system cannot mount a robust response even were it able to develop CD8 killer cells specific to the HIV epitope they vaccinate with.
Phase I trials are high risk, and especially for a bootstrapped team like this, have a lot riding on them. An early failure can doom a technology in this industry, so it is important to focus on testing in an environment where you have the best shot at success. Downstream studies can focus on other applications if needed.
Why would a company not test a technology that may well benefit both infected and uninfected people on both populations? You would only have more data. It will take much longer to see the results if you are only studying it as a preventative measure because you have to wait (a really long time, I would suspect) to see which of your study ends up getting HIV. Additionally, you'd need to study those that do get HIV (if some do, and some probably would, as even the best vaccines aren't 100% effective) and try to see if you can tell if their infection progresses differently or if it is somehow augmented by the vaccination's boost to the immune system. I mean... your explanation doesn't actually make that much sense to me.
Why not test both populations?