Paying the tribute of a sentimental sigh or two?

Hello everybody.
I'm reading 'Little Women' by Louisa May Alcott.

They all liked Jo immensely, but never fell in love with her, though very few escaped without paying the tribute of a sentimental sigh or two at Amy's shrine. And speaking of sentiment brings us very naturally to the 'Dovecote'.

Would you please tell me what the bold part means?

Does it mean very few left there without falling in love with Amey. In other words, many boys were fascinated by Amey and left there after they romantically sighed a bit Infront of her, am I right.

Thanks ??