I think you definitely have a very good go at this question you asked.
Overall, you are spot on. IP has an interesting liability profile as you have pointed out. But seeing the big picture, when you think about the investment strategy, you would have faced with a book of IP not just one policy.
Based on the target customer groups you are after, you should have a stable portfolio in terms of their mean duration of claim payout time. So you should be able to see that overall, your liability should be fairly stable over time, unless there is a company strategy to expand or shrink the book. As a result of this, you should hold investment like bonds, (both govt and corp, depending on the overall risk appetite of the company), which will give you stable income to match your liability. IL bonds would be good to have if some of your IP liabilities are IL.
Cash is essential to run a business to act as a buffer of fluctuation of claims payout.
IL investment might be essential for expense related items due to their nature.
Equities and properties might be good on the side. But the key thing is to ensure that all your liabilities are matched. Free assets may be used to invest in those categories of real assets. But in practice, the free assets may be invested in the same way as the liability-matching assets.