On a conjecture due to J.-L. Colliot-Thelene

Let $R$ be a regular local ring containing a field, $K$ be its fraction field, $a\in R^{\times}$ be a unit, $n\geq 1$ be an integer, $1/2$ is in $R$. Particularly, we prove the following result. Suppose a is a sum of n squares in K. Then a is a sum of n squares in R. This is a partial case of a conjecture due to J.-L. Colliot-Thelene (1979). The conjecture is solved in positive for regular local rings containing a field.
In more details. If R contains rational numbers, then the conjecture is solved by the speaker in his Inventiones paper (2009). If R contains a finite field and the residue field of R is infinite, then the conjecture is solved by the speaker jointly with K.Pimenov in 2010 in their Doc. Math. paper. If R contains a finite field, then the conjecture is solved by S. Scully in 2018 in his Proceedings of the AMS paper. If time permits, very recent progress in the topic will be discussed.