It seems that few people were interested in this problem. In the early days, the main focus was in experimentally testing Bell inequalities. The papers I know that treat the problem abstractly are from the 80s onward.

The story goes like this:

In 1969, Clauser, Horne, Shimony and Holt publish their famous paper that introduced the CHSH inequality. In it, they stated the the maximal violation for a singlet is $2\sqrt{2}$. As they were analysing a specific experiment, they didn't bother proving it for any two-qubit state, nor even stated the bound explicitly. They were inspired by Bell (of course) and Bohm's 1957 paper with Aharonov. Amusingly, Bohm's paper displays an inequality whose quantum bound is 2.85, and the bound for some local hidden-variable models he tried is 2. Unfortunately, these numbers are just a numerical coincidence, as his inequality has nothing to do with CHSH, not being even a Bell inequality.

In 1978, Boris Tsirelson gives a seminar on Bell inequalities. The chairman (A. Vershik) asks him if it's possible to develop analogue inequalities for quantum theory, i.e., to bound the strength of quantum correlations.

In 1980, Boris Tsirelson publishes a paper that proves that in the CHSH case, it is possible to bound the correlations by $2\sqrt{2}$, for any quantum state.

In 1985, Summers and Werner rediscover Tsirelson's bound independently. (Apparently it was they who popularised it outside the Soviet Union; Landau in 1987 cites both Tsirelson and Summers.)

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