Inequality $a^{\tan(b)}+b^{\tan(c)}+c^{\tan(a)}\leq 3(\frac{1}{3})^{\tan(\frac{1}{3})}$

Question

Hi I have a new problem this is the following :

Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$a^{\tan(b)}+b^{\tan(c)}+c^{\tan(a)}\leq 3(\frac{1}{3})^{\tan(\frac{1}{3})}$$

I know that $f(x)=x^{\tan(x)}$ is convex on the interval $[0;1]$ but I think it doesn't help at all .

I try also rearrangement inequality but it fails .

If you have hints it would be nice tahnks a lot .