I believe so, he has the duty to private "accredited" investors. IIRC, this is sort of what toppled Shkreli-- he mishandled investor money in a criminal manner, despite actually not losing any of their money.
Regardless of that execrable Shkreli, you have de jure fiduciary duties if you are on the board of a private company, yes. At least in the most common forms of incorporation. I am not even sure if it requires you have other shareholders, let alone if you are a controlling shareholder or they are accredited investors.
