Asymptotic order conjecture for higher xor-powers of Kneser graphs

Let $f_\ell(n,k)$ denote the clique number of the xor-product of $\ell$ copies of the Kneser graph $KG(n,k)$. Fix $\ell$ and suppose that $k$ is sufficiently large. Higher-power growth conjecture.

$$f_\ell(n,k)=\Theta\left(n^{\left\lfloor\log_2(\ell+1)\right\rfloor}\right).$$

The preceding bounds show the corresponding polynomial exponents in the relevant range and establish the exact magnitude for $\ell\leq 4$. The conjecture asserts that the displayed exponent gives the true order for every fixed $\ell$ when $k$ is large enough.