Eventual linear growth governed by elementary subalgebras

Determine whether the dimensions of higher commuting varieties exhibit eventual linear growth governed by varieties of elementary subalgebras.

Background

The introduction discusses higher commuting varieties of nilpotent matrices and notes that, for sufficiently large tuple length, their dimensions can be described by a linear-growth formula. In the cited work of Levy, Ngo, and Šivic, the eventual behavior is formulated as a conjecture in terms of parameter varieties of elementary subalgebras. The present paper develops an analogous characteristic-zero framework for symmetric pairs using varieties of abelian nilpotent subspaces and proves eventual linear growth in that setting, but the quoted conjecture is presented as an external unresolved prediction.