Asymptotics for number of indecomposable components of tensor powers of the natural $\mathrm{SL}_2$-module in odd characteristic

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $δ_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)