Zero Lebesgue measure of the Mandelbrot set boundary

Prove that the boundary of the Mandelbrot set has zero Lebesgue measure, including the infinitely renormalizable parameters with unbounded combinatorics that remain outside the cases addressed by existing results.

Background

The conjecture is presented as a probabilistic counterpart to the Mandelbrot local connectivity conjecture. The paper notes that zero area is known for the at-most-finitely-renormalizable portion of the boundary and for infinitely renormalizable parameters with bounded combinatorics.

According to the discussion, the remaining case is the set of infinitely renormalizable parameters with unbounded combinatorics. A complete renormalization theory for quadratic polynomials is identified as a likely prerequisite for resolving the problem.