Fractal behavior of tensor powers of tilting modules of $\text{SL}_2$

Given a group $G$ and $V$ a representation of $G$, denote the number of indecomposable summands of $V^{\otimes k}$ by $b_k^{G, V}$. Given a tilting representation $T$ of $\text{SL}_2(K)$ where $K=\overline{K}$ and of characteristic $p>2$, we show that $Ck^{-\alpha_p}(\text{dim} T)^k<b_k^{T, \text{SL}_2(K)}<Dk^{-\alpha_p}(\text{dim} T)^k$ for some $C, D>0$ where $\alpha_p=1-(1/2)\log_p(\frac{p+1}{2}).$