Peternell's algebraic approximation conjecture for minimal Kähler varieties

A Kähler variety here is a compact complex variety admitting a Kähler metric, and a minimal Kähler variety is one that cannot be blown down any further. An algebraic approximation is a deformation whose nearby member is algebraic.

Peternell's conjecture. Any minimal Kähler variety has an algebraic approximation.

If true, this would imply that every minimal compact Kähler manifold can be deformed into a projective manifold. The corresponding statement without minimality is false in dimensions at least four by examples of Voisin, while the three-dimensional Kodaira problem has an affirmative answer; the minimal version stated here remains open.