Problem is, hardly anyone will care about the open register, except to whale-watch and follow them.
(In AU, UK and EU you can replace "hardly anyone" with "nobody willing to speak")
Removing the loophole for insider trading would be a specific treatment, but only dismantling the power pyramid can cure it.
"When buying and selling is subject to legislation, the first things bought and sold are the legislators."
- Murray Rothbard, I think