As a matter of supposition; in the beliefs or according to the claims of some people.
People from other planets have supposedly visited Earth in flying saucers.
According to your testimony, you were supposedly at home watching TV when the murder occurred.
2011, Tom Fordyce, Rugby World Cup 2011: England 12-19 France 
France were supposedly a team in pieces, beaten by Tonga just a week ago and with coach Marc Lievremont publicly berating his players, but so clear-cut was their victory that much of the atmosphere had been sucked from the contest long before the end.