Asymptotic formula for linear-system chromatic number

For integers $k\ge 3$, let $f_1(e,q,k)$ denote the maximum chromatic number of a $1$-$(q,k)$-system with $e$ $q$-cliques. The asymptotic chromatic-number conjecture. Fix $k\ge 3$. There exists $C=C_k>0$ such that if $e>q^C$, then

$$f_1(e,q,k)=\Theta\!\left(e^{\frac{1}{2k-2}}q\right)\qquad(q\to\infty).$$

The conjecture predicts the sharp order of magnitude in the large-$e$ regime, improving the preceding upper and lower bounds. The source describes the problem as wide open for smaller values of $e$ and gives no resolution of this asserted asymptotic statement.