The geometric $P=W$ conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In a new version of a joint work with Enrica Mazzon and Matthew Stevenson, we prove the full geometric $P=W$ conjecture for elliptic curves: this is the first non-trivial evidence of the conjecture for compact Riemann surfaces. As a byproduct, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. We also explain how the geometric $P=W$ conjecture by Katzarkov-Noll-Pandit-Simpson is related to the cohomological version formulated by de Cataldo-Hausel-Migliorini.