A graph G is equitably k-list arborable if for any k-uniform list assignment L, there is an equitable L-colouring of G whose each colour class induces an acyclic graph. The smallest number k admitting such a coloring is named equitable list vertex arboricity and is denoted by ρ=(G). Zhang in 2016 posed the conjecture that if k (∆(G) + 1)/2 then G is equitably k-list arborable. We give some new tools that are helpful in determining values of k for which a general graph is equitably k-list arborable. We use them to prove the Zhang’s conjecture for d-dimensional grids where d ∈ {2, 3, 4} and give new bounds on ρ=(G) for general graphs and for d-dimensional grids with d ≥ 5.