Jannsen's vanishing conjecture for Hecke-character realizations

Let $K$ be the imaginary quadratic field and let $M_{\theta\mathbb{Q}_p}(n)$ be the $p$-adic realization of the Hecke-character motive defined in the source, with weight $w$. Let $p$ be a prime at which $E$ has good reduction at the primes over $p$ in $K$, and let $S$ contain the primes over $p$ and the primes dividing the conductor $\mathfrak{f}_{\theta}$. Jannsen's conjecture for Hecke characters. One should have

$$H^2(\operatorname{Gal}(K_S/K),M_{\theta\mathbb{Q}_p}(n))=0$$

for $n>w+1$ or $w+1>2n$. This is the specialization of Jannsen's conjecture to the concrete realizations attached to Hecke characters; the source presents it as a conjecture and does not establish it in general.