OLYMPIC-REVENGE 2021 Problem 1

Let $a$ , $b$ , $c$ , $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$ . Prove that:

$$\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.$$

Proposed by Zhang Yanzong and Song Qing