The inequality can be rewritten as follows:
$$\begin{aligned} & (x^4 + \frac{1}{z^4} + z^4 + \frac{1}{y^4}) + (y^4 + \frac{1}{t^4} + t^4 + \frac{1}{z^4}) + (z^4 + \frac{1}{x^4} + x^4 + \frac{1}{t^4}) + (t^4 + \frac{1}{y^4} + y^4 + \frac{1}{x^4}) \\ & \ge 4 \cdot \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{t} + \frac{t}{x} \right). \end{aligned}$$

$(x^4 + \frac{1}{z^4}) + (z^4 + \frac{1}{y^4}) \geq 2\sqrt{x^4 \cdot \frac{1}{z^4}} + 2\sqrt{z^4 \cdot \frac{1}{y^4}} = \frac{2x^2}{z^2} + \frac{2z^2}{y^2} \geq 4\sqrt{\frac{x^2}{z^2} \cdot \frac{z^2}{y^2}} = \frac{4x}{y z},$
that finishes the proof.