$\newcommand{\der}{\mathrm{der}}\newcommand{\tdert}{\mathrm{dert}}\newcommand{\erf}{\operatorname{erf}}\newcommand{\eqs}{\overset{\text{sign}}=}\newcommand{\tder}{\widetilde\der}$The answer is no.   

Indeed, let $x:=\xi_1$ and $y:=\xi_2$, so that $\xi_3=1-x-y$, $0\le x\le y\le1-x-y$, whence $x\in[0,1/3]$. 

Consider further the case $y=x\in(0,1/3]$, so that 
$$\mu_1=M(x):=\sqrt{x/2},$$
$$\rho=-R(x),\quad R(x):=\sqrt{\frac{1-2x}{2(1-x)}}.$$
Let also 
$$p_t(x):=1-P(X_1\le t,X_2\le t).$$
Your desired result would imply that 
\begin{equation*}
	p_t(x)\le p_t(1/3) \tag{0}
\end{equation*}
for all $x\in(0,1/3]$ and all real $t>0$. 

Note that 
\begin{equation*}
	p_t(x)=1-P(X_1\le t,X_2\le t)=P(M(x),-R(x)), \tag{1}
\end{equation*}
where 
\begin{equation*}
	P(m,r):=1-\int_{-\infty}^t du \int_{-\infty}^t dv\, f_{m,r}(u,v)  
\end{equation*}
and $f_{m,r}$ is the density function of the bivariate normal distribution with means $m,0$, variances $1,1$, and correlation $r$. 

The key is Plackett's observation ([formula (3)][1]) that 
\begin{equation*}
	D_r f_{m,r}(u,v)=D_v D_u f_{m,r}(u,v),
\end{equation*}
where $D_w$ denote the partial derivative with respect to a variable $w$. It follows that 
\begin{equation*}
\begin{aligned}
	D_r P(m,r)&:=-\int_{-\infty}^t du \int_{-\infty}^t dv \,	D_r f_{m,r}(u,v) \\ 
	&=-\int_{-\infty}^t du \int_{-\infty}^t dv \,	D_v D_u f_{m,r}(u,v) \\ 
	&=-f_{m,r}(t,t).  
\end{aligned} 
\tag{2}
\end{equation*}
Next,
\begin{equation*}
\begin{aligned}
P(m,r)&=1-\int_{-\infty}^t du \int_{-\infty}^t dv\, f_{0,r}(u-m,v)  \\ 
&=1-\int_{-\infty}^{t-m} dw \int_{-\infty}^t dv\, f_{0,r}(w,v) 
\end{aligned} 
\end{equation*}
and hence 
\begin{equation*}
\begin{aligned}
	D_m P(m,r)=\int_{-\infty}^t dv\, f_{0,r}(t-m,v)=\int_{-\infty}^t dv\, f_{m,r}(t,v);  
\end{aligned} 
\tag{3}
\end{equation*}
the latter integral can be easily expressed in terms of the error function $\erf$ and elementary functions. 

By (1) and a chain rule of differentiation, 
\begin{equation*}
	D_x p_t(x)=D_m P(M(x),-R(x))M'(x)-D_r P(M(x),-R(x))R'(x). 
\end{equation*}

In particular, 
\begin{equation}
	D_x p_{1/10}(x)\big|_{x=1/3}=\erf\left(\frac{1}{60} \left(3 \sqrt{6}-10\right)\right)+1-\frac{3
   }{\sqrt{\pi }}\,e^{(30 \sqrt{6}-77)/1800}
   =-0.739\ldots<0. 
\end{equation}
So, inequality (0) fails to hold for $t=1/10$ and all $x$ in a left neighborhood of $1/3$. $\quad\Box$

[1]: https://www.jstor.org/stable/2332716