Is it true that the smallest root $t$ of the polynomial

$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \cos \beta \cos \gamma - 1
$$

is always no greater than $\frac{1}{5}$ for all $\alpha, \beta, \gamma \in [0,\pi]$ with $\alpha + \beta \geq \gamma$, $\alpha + \gamma \geq \beta$, $\beta + \gamma \geq \alpha$, and $\alpha + \beta + \gamma \leq 2\pi$?

Thanks a lot for any helpful answer.