The non-abelian Euler-characteristic criterion for twisted $L$-values

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $F_\infty/\mathbb{Q}$ be the false Tate curve extension, and let $G=\operatorname{Gal}(F_\infty/\mathbb{Q})$. Let $\tau:G\to\operatorname{GL}_n(\mathbb{Q}_p)$ be self-dual, and let $\chi_{na}(E,\tau)$ denote the corresponding non-abelian Euler characteristic. Non-abelian Euler-characteristic conjecture.

$$L(E,\tau,1)\ne0\quad\Longleftrightarrow\quad \chi_{na}(E,\tau)\text{ is finite}.$$

When this holds, the non-commutative main-conjecture equality for $\tau$ should also hold. This connects nonvanishing of twisted complex $L$-values with finiteness of the twisted Selmer-theoretic Euler characteristic.