Well it's not just optimism that causes me to disbelieve the figure of 5% having been infected (or wasn't it just 1% according to data from the app that you subscribe to?), it's my gut instinct - or perhaps my 'good old British common sense'. The Cambridge modelling (in conjunction with Public Heath England) suggests that, at the peak, 213,000 people a day were contracting the virus in London. For the sake of argument, if we suppose that the two weeks leading up to that peak and the two weeks leading down from it averaged 100,000 per day, that would suggest 3 million infections in London in one month - 30% of the population.
These numbers ring true to me, and, given that they suggest that London could be Covid free by the end of the month, I think we should start making plans for the pub meet-up in June.