Bounds for SL₂-indecomposables in tensor powers of the natural representation in characteristic $2$

Let K be an algebraically closed field of characteristic $2$, G be the algebraic group SL₂ over K, and V be the natural representation of G. Let bₖG,V denote the number of G-indecomposable factors of V⊗ k, counted with multiplicity, and let δ = 32 - log 3/2log 2. Then there exists a smooth multiplicatively periodic function ω(x) such that b₂ₖG,V = b₂ₖ₊₁G,V is asymptotic to ω(k) k-δ4ᵏ. We also prove a lower bound of the form cW k-δ( W)ᵏ for bₖG,W for any tilting representation W of G.