Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$=\{ 0,1 \}, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$=\{$\xi<\mathfrak{c}:y(\xi)=1$\}, the *support* of $y$, and let $X$=\{$x\in Y:0<|\operatorname{supp}(x)|\le\omega_1$\}; $|X|=\mathfrak{c}^{\omega_1}=(2^{\omega_1})^{\omega_1}=\mathfrak{c}$. 

Does X have $G_\delta$ diagonal?