Necessary and sufficient conditions for scatteredness of the generalized quadrinomial

Let $t\geq 3$, let $1\leq s\leq 2t-1$, and let $m\in\mathbb F_{q^t}$ and $h\in\mathbb F_{q^{2t}}$. Consider the $q^s$-polynomial

$$\psi_{m,h,s}=m\left(X^{q^s}-h^{1-q^{s(t-1)}}X^{q^{s(t-1)}}\right)+X^{q^{s(t+1)}}+h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}\in\widetilde{\mathcal L}_{n,q,s}.$$

Scatteredness conjecture. The polynomial $\psi_{m,h,s}$ is scattered if and only if $m$ and $h$ satisfy the assumptions in Theorem~.

The conjecture asserts that the sufficient conditions established in Theorem~ are also necessary. The paper reports computational evidence for this necessity, while the general equivalence remains open.