Any tricks to make this background true white?

Well, in that case, you break out the lasso or paint those in black on that black and white layer. Or you set the magic wand to only accept contiguous areas when doing the selection.

If you just want to color balance the white background, you can copy the b&W layer and use it as the mask for a application of levels. Just copy it into the layer mask should work. And use that grey eyedropper to balance the color or move the middle triangle till you like the result.