Standard conjecture on u-invariants?

Question

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin (in Fields of $u$-invariant $9$) in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

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¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL review.

Answer 1

For the classical $u$-invariant of fields of characteristic $\neq 2$, some known results are:

1. The $u$-invariant of formally real fields is $\infty$.

2. If $K$ is an algebraically closed field, its $u$-invariant is $1$. More generally, if $K$ does not have quadratic extensions its $u$-invariant is $1$.

3. There are no fields of $u$-invariant $3, 5$ or $7$. This was proven by R. Elman, T. Y. Lam, ( Math. Z. 131 (1973), 283--304 and Invent. Math. 21 (1973), 125--137.)

4. There are fields of any even $u$-invariant (A. S. Merkurev, Izv. Akad. Nauk. SSSR Ser. Mat. 55 (1991) No. 1, 218--224.)

5. There is a field of $u$-invariant $9$ (O. T. Izhboldin, Ann. of Math. (2) 154 (2001), no. 3, 529--587)

6. There is a field of $u$-invariant $2^r+1$ for every integer $r\geq 3$ (A. Vishnik, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 661--685, Progr. Math., 270, Birkhäuser Boston, Boston, MA, 2009.)

On the other hand, for certain families of fields, the $u$-invariant is known, and for other families, the computations are not complete. For example, if $F$ is a finite field of odd characteristic by the Chevalley-Waring theorem every quadratic form in three variables over $F$ represents $0$, that is, every $3$-dimensional form over $F$ is isotropic and thus $u(F)=2$.