I’m not a big fan of “the little white lie” but in this case, what they don’t know, won’t hurt you. I’ve had too many clients tell people they were injured, then when they got better, people were concerned that they would be a liability. If you trust the people completely, then tell them the truth. But why would you be asking me if you trusted them completely.
PS word spreads fast! And the grapevine tends to exaggerate as it grows.