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

Let $a$, $b$, $p_1$, and $p_2$ be nonzero constants, and consider the differential equation

$$f^2(z)+a(f'(z))^2=p_1e^{bz}+p_2e^{-bz}.$$

Gao's conjecture. If $9ab^2=-4$, then every entire solution is either

$$f(z)=l_1e^{\frac{-3}{2}bz}+l_2e^{\frac{1}{2}bz},$$

where $l_2^2=\frac{9}{8}p_1$ and $l_1^2=\frac{p_2^2}{8p_1}$, or

$$f(z)=l_3e^{\frac{3}{2}bz}+l_4e^{\frac{-1}{2}bz},$$

where $l_4^2=\frac{9}{8}p_2$ and $l_3^2=\frac{p_1^2}{8p_2}$.

The conjecture concerns the exceptional parameter case excluded from the preceding classification theorem. It is motivated by the explicit solution $f(z)=e^z+e^{-3z}$ when $a=-\frac19$ and $b=2$; the source does not state whether the conjecture has been resolved.