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 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? Or is it completely open, with no prevailing hypothesis?