But how would you tackle it without making the problem worse?
That's what OP means about there not being an incentive for the government to lower cost of living. The high cost of living is subsidising HK's notoriously low taxes.
There's a pretty good YouTube video about it: https://www.youtube.com/watch?v=hLrFyjGZ9NU
Oh and to answer the question, the solution is to zone lots more high-density residential, sell a lot of land for less money, and stop subversively taxing the poorest citizens via high CoL. Tax businesses and the wealthy to make up for it. But they're not going to do that, because low taxes on businesses and the wealthy is their USP.
Turns out when you optimise a city for business wellbeing at the expense of citizen wellbeing, then it impacts citizen wellbeing...
In the process, it makes land completely unappealing to rent-seekers while incentivizing high-density development of desirable locations.
This is only a problem if you believe older folks should continue to live in big luxury houses for a pittance (read: gain the benefits of luxury without having the detriments of luxury), or the country does not have enough smaller housing for them to move.
In situations where housing is severely constrained at the cost of younger people, leaving old people in their large houses should be the last of your concerns. Doubly so for any social structures where older generations are extremely reliant on younger generations (almost every democratic, socialist or communist country).