Noncommutative root stacks?

Seeing noncommutative projective geometry and root stacks in one installment of fortnightly links, I'm led to the question whether it makes sense to consider root stacks of noncommutative projective planes (and more general noncommutative varieties, under suitable conditions). These all have a "commutative divisor", and given that one can blow up noncommutative surfaces (à la Van den Bergh) in points on this divisor, maybe root stacks also have an incarnation in this setting.

When the input is a homogeneous coordinate ring together with a normal element such that the quotient ring describes the commutative divisor, I would hope it is possible to define a $\mathbb{Z}\oplus\mathbb{Z}$-graded ring (à la the scaled Rees rings discussed by Van Oystaeyen) so that a suitable qgr gives the abelian category of the root stack. Its derived category should have the expected semiorthogonal decomposition, which for blowups is in fact known.

If the noncommutative projective plane is finite over its center I've been expecting for a few years that this construction (or something similar) leads to something interesting but I haven't worked out the details unfortunately. But I now hope some construction might work in sweeping generality.

Let me know if you are interested in this!