Is the following claim valid?

**Claim:** Given any planar convex region C, the thinnest cover of the plane with copies of C cannot have any region where more than 2 copies overlap. In general, the thinnest n-fold cover of the plane with copies of C cannot have regions where more than n+1 copies overlap. Copies of C are allowed to be rotated.

Note: The question can be asked in higher dimensions and in non-Euclidean setting. If C is allowed to be non-convex, the thinnest cover of the plane with C units *might* have regions where arbitrarily large number of units overlap.

A related issue was raised in On Covering a Planar Region with Copies of a Tile of Different Shape

A very elegant 2-fold covering of the plane is shown at: Thinnest 2-fold coverings of the plane by congruent convex shapes