will a large roll fit in the recess?

Yes. We regularly install what manufacturers claim to be "Mega-rolls" with no problem on the Moen Lounge Spot Resist Brushed Nickel Surface Mount Toilet Paper Holder, Lowes Item #: 176885 | Model #: DN7708BN. When you install the holder, remember to leave enough space between the end of the holder's rod (the trumpet end) and any adjoining walls, toilet, etc so you have room to get the large roll onto the rod. If the holder is installed correctly, the rod does not swing freely.