Gao's conjecture on entire solutions of a Fermat-type differential equation

Let $l$ and $n$ be positive integers with $n>4$, and let $p_1$, $p_2$, $a$, $a_1$, and $a_2$ be nonzero constants, with $a_1\ne a_2$. Consider the differential equation

$$f^n(z)+a(f^{(l)}(z))^n=p_1e^{a_1z}+p_2e^{a_2z}.$$

Gao's conjecture. This equation has no entire solution.

The conjecture is a nonexistence claim for higher powers of a derivative term with two exponential terms on the right-hand side. The paper states that its Corollary 1 gives a positive answer to this conjecture; consequently, the conjecture is solved.