The Malle–Türkelli bounds for number fields

Let $k$ be a number field and let $G\subset S_n$ be a transitive permutation group. Let $b_M(G,k)$ be Malle's logarithmic constant, $b(G,k)$ the actual logarithmic constant, and $b_T(G,k)$ Türkelli's modified constant. The Malle–Türkelli bounds. Then

$$b_M(G,k)\leq b(G,k)\leq b_T(G,k).$$

The inequalities would place the actual exponent between the original and modified predictions. The paper presents examples motivating them, but does not establish the assertion in general.