Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by
$$F(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$
For every $a\in [0,1)$, does there exist $C\equiv C_a>0$ s.t.
$$\left|\int_0^\infty \frac{1}{s-t}\left(F\left(\frac{y}{\sqrt{c(s-t)}}\right)-F\left(\frac{y-x}{\sqrt{c(s-t)}}\right)\right) dy\right| \le \frac{C|x|^a}{(s-t)^{(1+a)/2}}$$
holds for all $0\le t<s\le T$ and $x\in\mathbb R$?