So I got a request from a friend to try and solve this, but I've stuck.
$$\alpha^\alpha\left(1 - \alpha \right)^{1-\alpha} \geq \beta^\alpha \left(1 - \beta\right)^{1-\alpha}$$
for $\alpha, \beta \in (0,1)$.
I have reached here:
$$\left(\frac{\alpha}{\beta}\right)^\alpha\geq\left(\frac{1-\beta}{1-\alpha}\right)^{1-\alpha}$$
but maybe I have to use a theorem or an already proven inequality that I can't seem to come up to. I thought trying first $0\lt\alpha\lt\beta\lt1$ and then $0\lt\beta\lt\alpha\lt1$ but can't find my way through.
We need to prove that: $$\alpha\ln\alpha+(1-\alpha)\ln(1-\alpha)\geq\alpha\ln\beta+(1-\alpha)\ln(1-\beta).$$ Consider $f(\beta)=\alpha\ln\beta+(1-\alpha)\ln(1-\beta).$
Thus, $$f'(\beta)=\frac{\alpha}{\beta}-\frac{1-a}{1-\beta}=\frac{\alpha-\beta}{\beta(1-\beta)},$$ which gives $\beta_{\max}=a$ and since $$f(\alpha)=\alpha\ln\alpha+(1-\alpha)\ln(1-\alpha),$$ we are done!