Positive rank over the eighth cyclotomic field

Let $E/\mathbb Q$ be an elliptic curve with split multiplicative reduction at $2$. Cyclotomic positive-rank conjecture. The curve has infinitely many rational points over $\mathbb Q(\zeta_8)$, equivalently

$$\operatorname{rk}E/\mathbb Q(\zeta_8)>0.$$

The source presents this as a consequence predicted by the parity conjecture, using the same splitting ideas as for the field $\mathbb Q(i,\sqrt{17})$. It remains conditional on parity.