Counterexamples to Magnanini's conjecture concerning critical points of the torsion function

We construct planar simply-connected domains whose torsion function has an arbitrary prescribed number of local maxima while the distance to the boundary has only one local maximum inside the domain. This disproves a conjecture of Magnanini asserting that the number of local maxima of the torsion function is bounded above by the number of local maxima of the distance function to the boundary. We also answer a question of Steinerberger concerning the eccentricity of level sets near the point where the torsion function attains its maximum in simply-connected domains.