Janice's notes confirm the link -- the Honest John's at 3rd & Ogden became the early Lady Luck.

On close inspection of the 2 photos, the "Money Orders" sign is very similar and the "Payroll Checks Cashed" sign is identical.
There's other bits and pieces that Lady Luck retained as well.

I've also got a series of annual reports from Lady Luck Corporation when it was a public company. In one of them, they trace their beginnings and I think it included reference to the Honest John's location. I'd have to dig quite a bit to find it though.