If mortgages increase demand for home-buying and therefore increases price, do reverse mortgages imply the opposite?

i.e. price reduction due to increased supply

My logic: if a housing market moves from cash only buyers to buying with a mortgage, the demand for houses will increase as there will now be more buyers. Therefore, prices go up. This is what we see every time a new policy is enacted that helps buyer (eg. help to buy in the UK).

Now I'm struggling to contemplate the opposite. At the moment reverse mortgages are not very popular, partly for political reasons – you die and the bank keeps the house, leaving nothing for your kids to inherit. But let's say suddenly every homeowner with a fully paid off house takes out a reverse mortgage.

First of all, the bank would have to assess the current value of the property to see how much they could take out. But because you're not selling the house (yet), there's no increase in supply, so prices shouldn't drop. But once you die or the reverse mortgage ends, you have to pay back what you borrowed, which will most likely mean to sell the house. At this point i can see prices going down – but only in the long term.

Is this an effect that has been studied or talked about that I could read more into?