Inequality in 3 variables (conjecture)

Question

Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$. If $0<k\leq 3+2\sqrt{3}$, then $$\frac{a}{b^2+k}+\frac{b}{c^2+k}+\frac{c}{a^2+k}\geq \frac{3}{1+k}$$ If $k=3+2\sqrt{3}$, then equality occurs if $a=b=c=1$ or $a=0$, $b=1-\sqrt{3}$ and $c=\sqrt{3}$ or any cyclic permutation thereof. This inequality can be found in Vasile Cirtoaje's Discrete inequalities, Volume 4 - page 61. Any idea for this inequality? Thank you!