Symmetric powers of Ramanujan tau-function Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.
For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$ is conjectured to be holomorphic and satisfies a functional equation. 
Question 1: For which $k$, has $Sym^k\Delta$ been proved to be automorphic form on $\mathrm{GL}(k+1)$? 
Question 2: For which $k$, has $L(s,Sym^k\Delta)$ been proved to be homorphic and satisfies a functional equation?
I know certainly that both questions is YES for $k=2,3,4$ (Gelbart-Jacquet, Kim-Shahidi, Shahidi). But I heard there is progress beyond that because $\Delta$ is a modular form rather than arbitrary automorphic representation on $\mathrm{GL}(2)$. 
 A: It is indeed true that substantially more is known for holomorphic cusp forms than for general automorphic representations, as a consequence of modularity lifting theorems. 
The strongest result so far seems to be due to Clozel and Thorne, who have proved automorphy of $Sym^k(f)$, for $f$ a holomorphic modular form of weight $\ge 2$, for all $k \le 8$. This answers your Questions 1 and 2 for these $k$.
I'm not aware of any cases where Question 2 is known without also answering Question 1 (assuming you want holomorphy on all of $\mathbf{C}$, rather than some smaller domain). But if you're willing to settle for $L(s, Sym^k \Delta)$ being meromorphic on $\mathbf{C}$ with the expected functional equation, then this is known for all $k$, as a consequence of the potential automorphy theorems of Barnet-Lamb, Geraghty, Harris and Taylor. This is discussed in Clozel and Thorne's article.
A: According to this exciting new preprint of Newton and Thorne (https://arxiv.org/abs/1912.11261), the answer to both Questions 1 and 2 is "all $k\geq 1$".
