Brück's conjecture for entire functions sharing a finite value with their derivative

Let $f$ be a non-constant entire function. Its hyper-order is denoted by $\rho_1(f)$; two meromorphic functions share a value $a$ CM when their $a$-points, counted with multiplicities, coincide.

Brück's conjecture. If

$$\rho_1(f)\notin\mathbb{N}\cup\{\infty\},$$

and $f$ and $f'$ share one finite value $a$ CM, then

$$f'-a=c(f-a),\qquad c\in\mathbb{C}\backslash\{0\}.$$

This is a value-sharing analogue of uniqueness results for entire functions and their derivatives. The supplied text identifies it as a conjecture proposed by Brück, but gives no resolution evidence.