I don't think we will ever be able to prove FS didn't have a baby. There are too many possibilities.
As I said with my Grandma, even a miscarriage as early as 4 months can give you the same scar on the inside of the uterus (that shows up in an exam) a full term baby does.
She could also have thrown the baby in the trash

as desperate girls still do sometimes today, and then denied it ever existed.
She could have hidden from her family and friends while she was big and pregnant, then gave the baby to an orphanage (like AA said she did) and never mentioned it to anyone. The baby could have been born dead and she didn't tell anyone.
Also, her family and friends (at the time) could have known about it but kept her secret, (remember an 'out of wedlock' child was considered a shameful thing back then) and now they're dead and can't tell us.
So technically unless it was recorded at a medical office or a records builiding we cannot find evidence, and considering all the things that may have occured (especially during the turmoil of the WWI years) we'll never have proof. While you can prove someone did have a child, it's impossible to prove they
didn't unless they are still alive to be examined.