Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that $$ \operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\operatorname{dim}{H^0(G_k,V)}. $$

Beilinson has similar conjectures relating the LHS to algebraic K-theory rather than Selmer groups.

If $E$ is an elliptic curve, and we set $V=h_1(E)=h^1(E)(1)$, then the conjecture above is equivalent to the statement that the analytic rank is the same as the rank of the $p$-adic Selmer group. Beilinson's conjecture in this case is equivalent to the assertion that the analytic rank equals the Mordell-Weil rank.

For $V$ of non-negative weight, the conjecture simply asserts that the Selmer group vanishes. Assuming the conjectured properties of $L$-functions, the conjectures for $V$ and $V^*(1)$ are equivalent (in particular, the case of weight $\le -2$ follows from the case of weight $\ge 0$).

My question is: which cases of this conjecture are known for $V=\operatorname{Sym}^k{h^1(E)}(n)$ for $E$ an elliptic curve? I know that many cases are known when $k=n=1$ due to the theory of Heegner points, Gross-Zagier, etc, but I'd like to know what's known outside that range. (I'm especially interested in cases where $w=k-2n=-2, -3, -4$).

Feel free to give reference, or even better, specific elliptic curves in LMFDB.