I'll use this post to try and get a case together. Sometimes proving a negative is harder than proving a positive and I feel that the burden of saying this is Wilkinson should be on the people that are claiming it. Especially since I don't think it even looks like him. But, that being said, I'll do what I can.
In August 12, 1909 edition of the Humbolt Independent (Humboldt, Iowa), here's an article that seems to be talking about Hopkins Bros men's team. There is no mention of women and there would be if it was a novelty (which it was). Also no mention of Wilkinson. So we might assume that Wilkinson was gone by this time? I think it would be useful to try and get all of the Hopkins Bros Lady postcards that have postmarks to try and narrow down the dating of them.
I'm going to try and find a paper with Wilkinson managing the ladies team and get a date for that. But I still don't think that will prove anything. Even if I find an article with him managing the ladies team during the right time frame, that doesn't automatically put him in the picture.
Here's something interesting. They call the manager "Buster" Wilkinson. Also saying he was managing the Des Moines team and then is going to start managing the Ladies Road Team. So, that to me means that he coached both. Also puts a narrow window on his time with the Bloomer Team. April 20th, Summer of 1909. Still need to date the postcard.
