I think they should be able to own an investment portfolio, but while they are in office (and perhaps after, if they engage in lobbying) it should be in a blind trust.
I don't know details but in general the portfolio would be managed by a trustee, probably with a certain goal specified in advance (preservation of capital, growth, blended, etc.) the key is that the owner of the assets has no control over or knowledge of the specific investments once the trust is formed.
I don't know if a rollover into a blind trust has to involve a complete liquidation to cash to start off. That would make sense, since then there could be no assumptions about which securities the trust held.