Türkelli's modification of Malle's conjecture

Let $G\subset S_n$ be a transitive permutation group and let $Q$ be a global field with $(|G|,\operatorname{char}(Q))=1$. Let $b(G,Q)$ denote the relevant logarithmic exponent and let $b_T(G,Q)$ be Türkelli's modified constant. Türkelli's modification. One has

$$b(G,Q)=b_T(G,Q).$$

The modification was introduced to accommodate cyclotomic intermediate extensions that can invalidate Malle's original logarithmic exponent. The paper's counterexamples show that this modified prediction is also false in general.