Origin of "on the up-and-up"

The phrase "on the up-and-up" means "legitimate" or "legal" or "reputable" or, to use another idiom, "above-board". For instance:

Although Pete didn't look like a city official, Joe assumed his offer to sell the Brooklyn Bridge was totally on the up-and-up.

Where does this expression come from?

Related: What does “We don’t do anything that’s not completely up and up” mean?

Apparently its origin isn't clear. Grammarist.com reports:

The idiom on the up and up­­—whose exact origins are mysterious, though it dates from the late 19th century, is likely American, and appears to come from sports betting—means (1) open and honest, legitimate; and (2) on the rise. Though some dictionaries promote one or the other definition (usually the first), both senses are so widespread that we have to accept them.

The article then provides three citations for the first meaning, dated 1929, 1937, 1938, and one 1949 citation for the latter meaning. Merriam-webster.com says “First Known Use of UP-AND-UP 1863” for the meaning “an honest or respectable course”, but gives no citations. Languagehat.com has numerous comments on the two different meanings, but nothing about etymology; and Geoffrey Nunberg uses “on the up and up” as an example of a phrase that many people use one way, and many another, with people in either group not knowing of the other.

My grandpa once told me that the saying dates back to the railroad industry and union era in the United States. Railroad workers didn't have stable work, and so they were often switching companies or unable to get new work contracts. In addition workers in unions were more expensive, so some companies started marking union workers by making letters of recommendation such that the watermark was upside down. I'm not sure on the details and I'm sure it could've gone both ways, but according to my grandpa being on the up-and-up (watermark and text both upright) meant that you were going to get hired.