If every vegan turned 5 other people over the course of their lifetime, then we would only expect the number of vegans to grow something like 10x over a whole lifetime - 60 years or so - and the number of vegans has in fact grown enormously since the 1970s. Besides, many vegans who are included in surveys do not intend to stick with their diet, or do not want to seem "preachy," and so have less of an effect on their immediate circle.

However, population-level data on vegans is pretty spotty and you're right that I don't have any good evidence besides the Veganuary study cited in the piece, which only asked how many people surrounding the subject had tried veganism in a period of six months after the trial period.

The claim is primarily based on (1) my own experience, having been vegan for three years and having so far turned two close friends vegan and one vegetarian (2) my experience with professional activists, who can often turn one person vegan every week just by tabling for an hour or two. If you speak to other serious vegans (such as the commenter below who mentioned "several people") I suspect that they'll give you a similar number.