Small rational-object equality conjecture for finite groups

Let $G$ be a finite group. Let $f(G)$ denote the number of irreducible characters of $G$ with rational fields of values, and let $h(G)$ denote the number of conjugacy classes of $G$ with rational fields of values. Small rational-object equality conjecture. If

$$\min\{f(G),h(G)\}\leq 5,$$

then

$$f(G)=h(G).$$

The conjecture is motivated by computations of groups of order at most $128$, which found no counterexample with either $f(G)\leq 5$ or $h(G)\leq 5$. Whether the equality always holds under this bound remains open.