Given that, $x, y, z \in \mathbb{R}^{+}$
- $y – z = z- y \quad \longrightarrow (1)$
- $xyz = 4 \quad \longrightarrow (2)$
From equation $(1),$
$\Rightarrow y – x = z – y$
$\Rightarrow \boxed{2y = z + x}$
So, $x,y,z$ are in arithmetic progression.
We know that, $\text{AM} \geq \text{GM}$
$\Rightarrow \frac{x+y+z}{3} \geq \sqrt[3]{xyz}$
$\Rightarrow \frac{y+2y}{3} \geq (4)^{\frac{1}{3}}$
$\Rightarrow \frac{3y}{3} \geq (4)^{\frac{1}{3}}$
$\Rightarrow y \geq (2^{2})^{\frac{1}{3}}$
$\Rightarrow \boxed{y \geq (2)^{\frac{2}{3}}}$
Correct Answer $: \text{B}$