Let $\alpha+\beta+\gamma=\frac{\pi}{2}$ and $t>0$, prove that
$$\cos^t{(\alpha)}\sin{(\alpha-\beta)}\sin{(\alpha-\gamma)}+\cos^t{(\beta)}\sin{(\beta-\alpha)}\sin{(\beta-\gamma)}+\cos^t{(\gamma)}\sin{(\gamma-\alpha)}\sin{(\gamma-\beta)}\ge 0.$$
This result (which is actually a conjecture) has been suggested by equations $(14)$ and $(15)$ here.
Note. I am not an expert in inequalities, so this is probably known, obvious, or can be further generalized.
If $\alpha, \beta, \gamma \ge 0$:
WLOG, assume that $\alpha \ge \beta \ge \gamma$. Let $x := \alpha - \beta$ and $y := \beta - \gamma$. Then $x, y \ge 0$ and $x + y \le \pi/2$ and $\alpha = \gamma + x + y, \beta = \gamma + y$.
The desired inequality is written as $$\cos^t(\gamma + x + y)\sin x\, \sin (x + y) - \cos^t(\gamma + y) \sin x\, \sin y + \cos^t \gamma\, \sin(x + y)\,\sin y \ge 0 $$ or $$\cos^t(\gamma + x + y)\sin x\, \sin (x + y) + [\cos^t \gamma\, \sin(x + y) - \cos^t(\gamma + y) \sin x]\sin y \ge 0$$ which is true since $\cos^t \gamma \ge \cos^t(\gamma + y)$ and $\sin (x+y)\ge \sin x$.
