Zhang et al.'s exponential-form conjecture for difference sharing

Let $f$ be a transcendental entire function with order and exponent of convergence satisfying

$$\rho(f)<2,\qquad \lambda(f)<\rho(f).$$

For a positive integer $n$, let $\Delta^n f$ denote the $n$th forward difference with unit step, and let two functions share $0$ CM when their zeros coincide counting multiplicities.

Zhang et al.'s conjecture. If $f$ and $\Delta^n f$ share $0$ CM, then $f$ has the form

$$f(z)=ce^{dz},\qquad c,d\in\mathbb{C}\backslash\{0\}.$$

The conjecture was proposed after Zhang et al. questioned whether the nonzero auxiliary-function assumption in their partial result was necessary. The supplied text gives that motivation but no resolution evidence.