Conjecture on norm classes of maximum scattered subspaces in the new family

Let $q$ be a prime power and let $b_1,b_2 \in \mathbb{F}_{q^6}$. Define

$$f_i(x)=b_i x^{q}+x^{q^4}\in \mathbb{F}_{q^6}[x],\qquad i=1,2.$$

Assume that $f_i$ defines a maximum scattered $\mathbb{F}_q$-space $U_{b_i,1}$, and write $\operatorname{N}_{q^6/q^3}$ for the norm from $\mathbb{F}_{q^6}$ to $\mathbb{F}_{q^3}$. The preceding equivalence result identifies subspaces with equal norms.

Norm-class conjecture. The set

$$\left\{\operatorname{N}_{q^6/q^3}(b): f(x)=bx^{q}+x^{q^4}\text{ defines a maximum scattered }\mathbb{F}_q\text{-space }U_{b,1}\right\}$$

has size

$$\left\lfloor\frac{(q^2+q+1)(q-2)}{2}\right\rfloor.$$

If true, this would yield further examples of maximum scattered subspaces in the family. The conjecture was verified computationally with GAP for $q\leq 32$; no general proof or disproof is given.