Hodge standard conjecture for étale cohomology

I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic.

The proof in characteristic zero simply says that it is enough to consider $\mathbb{C}$ (via Lefschetz principle), and that there you can use Hodge theory, the Hodge index theorem in particular (via the comparison theorem).

Note that in positive characteristic the conjecture is known for surfaces, as it is a simple application of Riemann-Roch (see chapter V of Hartshorne for example). A proper answer should adress why no generalization of Riemann-Roch yields the result for other varieties.

Another way to phrase the main obstacle in positive characteristic is the following:

Although we can formulate the Hodge standard conjecture purely cycle-theoretically, in practice the only way we get a grip on it is through a cohomological representation. For example, in characteristic $0$ we use Hodge cohomology to prove the positivity statement of the pairing on primitive cohomology.

However, in positive characteristic, there isn't even a cohomology theory taking values in a field that admits a notion of positivity! The only known cohomology theories are over fields like $\mathbb Q_\ell$ and $W(k)$, neither of which admits the structure of an ordered field (because they have too many roots of unity).

Thus, one either has to come up with a cycle-theoretic proof that doesn't make reference to any cohomology theory (as far as I can tell, this would be new even in characteristic $0$), or one has to come up with a Weil cohomology theory taking values in an ordered field (like $\mathbb Q$ or $\mathbb R$, although already over $\mathbb F_{p^2}$ cohomology theories with values in one of these two fields provably do not exist by a result of Serre).

Edit: As suggested by Will Sawin below (Serre's original argument is the case $K = \mathbb R$):

Claim. There is no Weil cohomology theory with values in any totally ordered field $K$.

Indeed, there exists a supersingular elliptic curve $E$ over $\mathbb F_{p^2}$ such that $D = \operatorname{End}(E) \otimes_\mathbb Z \mathbb Q$ is a quaternion division algebra. It is non-split at $\infty$ (e.g. because it is nontrivial but split at all $v \neq p, \infty$), so it is given by $D = (a,b)$ for $a, b \in \mathbb Q_{<0}$. Such a division algebra never splits over an ordered field (since $a$ and $b$ have to remain negative in $K$, so $b$ can never be a norm of $K(\sqrt{a})/K$). $\square$

(In fact, using the Brauer–Hasse–Noether sequence from class field theory and explicit computation of the local invariant maps, one sees that $D$ can be represented as $$D = \left\{\begin{array}{ll}(-1,-1) & p = 2,\\(-1,-p) & p \equiv 3 \pmod{4},\\(-q,-p) & p \equiv 1 \pmod{4},\end{array}\right.$$ where in the third case, $q \equiv 3 \pmod{4}$ is a prime whose residue class mod $p$ is a non-square. However, we only needed the negativity of $a$ and $b$.)