Borel n-point sets in the plane for fixed n ≥ 2

Determine, for each fixed integer n ≥ 2, whether there exists a Borel subset M ⊆ R^2 that intersects every line l in exactly n points.

Background

Mazurkiewicz (two-point) sets are classical choice-based constructions where a set meets every line in a prescribed number of points. Mauldin studied the Borel feasibility when the number of intersection points varies with the line, showing the function l ↦ α_l must be Borel.

While there are simple Fσ examples for countably many intersection points, the case of a fixed finite number n ≥ 2 of intersection points on every line remains unresolved in the Borel context.