Li and Yang's exponential-type conjecture for entire solutions

Let $k$ be a positive integer, let $a,b\in\mathbb{C}$ with $a\neq b$, let $a_i\in\mathbb{C}$ for $i=0,1,\ldots,k$ with $a_k=1$, and let $\varphi$ be an entire function. For a function $f$, define

$$\mathscr{L}_k(f)=\sum_{i=0}^k a_i f^{(i)}.$$

Consider the algebraic differential equation

$$f^{(1)}\bigl(f-\mathscr{L}_k(f)\bigr)-\varphi(f-a)(f-b)=0.$$

When $\mathscr{L}_k(f)=f^{(k)}$, Li and Yang's conjecture. the entire solutions of this equation are functions of exponential type.

This conjecture concerns growth restrictions for entire solutions of a special algebraic differential equation. The corresponding case $\mathscr{L}_k(f)=f^{(1)}$ has been solved, but the general case with $\mathscr{L}_k(f)=f^{(k)}$ remains open.