Liu–Laine difference-sharing conjecture for entire functions

Let $f$ be a transcendental entire function of finite order satisfying

$$1<\rho(f)<\infty,$$

and let $c$ be a non-period of $f$. For a positive integer $n$, let

$$\Delta_c^n f(z)=\sum_{j=0}^n(-1)^{n-j}C_n^j f(z+jc)$$

be the $n$th forward difference. Two functions share a value $a$ CM when their $a$-points coincide counting multiplicities.

Liu–Laine's conjecture. If $f$ and $\Delta_c^n f$ share $a\in\mathbb{C}\backslash\{0\}$ CM, then

$$\Delta_c^n f-a=\tau(f-a),\qquad \tau\in\mathbb{C}\backslash\{0\}.$$

The conjecture was motivated by examples showing constant-ratio behavior for difference operators and by a theorem proving an analogous result when $\rho(f)<2$. The supplied text records no resolution of the stated finite-order range.