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](https://en.wikipedia.org/wiki/U-invariant).
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 $9$.<sup>1</sup>

> ***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.

<hr />
<sup>1</sup>
"Oleg Izhboldin died tragically ... at the age of 37 after submitting this article"
<a href="https://zbmath.org/?format=complete&q=an:0998.11015">ZBL</a>.