The publication is made by the UK institute of actuaries so I'm answering from the perspective of the insurance industry in the European union.
Is it allowed?
For European Insurance companies EIOPA gathers a number of statistics, among which asset exposures. The table below shows figures as of 2019Q4:
So, at least for insurance companies that are supervised by EIOPA and related supervisors, it's not true they are not allowed to hold equities.
Rules under Solvency II
European insurance companies are supervised under the Solvency II framework. Under this framework insurance companies need to ensure that they have enough capital to withstand once in 200 year shock using their own funds (basically 99.5% VaR), this is the Solvency Capital Requirement (SCR). The Dutch supervisor DNB defines basic own fund as
Basic own funds
The basic own funds consist of (i) the excess of assets over liabilities, and (ii) subordinated liabilities.
Eligible own funds
The classification into tiers is relevant to the determination of eligible own funds. These are the own funds that are eligible for covering the regulatory capital requirements – the solvency capital requirement and the minimum capital requirement. For example, the minimum capital requirement must be covered by Tier 1 and Tier 2 capital and may not therefore be covered by Tier 3 capital. The extent to which the tiers are eligible to cover the capital requirements is set out in the implementing measures (also known as delegated acts).
Not all own funds are eligible to cover the loss under the shock, eligible own funds made of high quality assets such as government bonds. The rest of the funds are under less strict rules. The Solvency II framework gives a detailed prescription of what funds must be held and how much risk stems for equity holdings and this impact the SCR calculation. The rules are quite detailed and far from a complete ban on equity holdings.