Uniform boundedness conjecture for ranks of quadratic twists

Let $E$ be an elliptic curve over a number field $k$, and call an elliptic curve $E'$ over $k$ a quadratic twist of $E$ when it is obtained from $E$ by a quadratic twisting operation over $k$.

Uniform boundedness conjecture for ranks of quadratic twists. There is a positive integer $R=R(E,k)$ such that every elliptic curve $E'$ over $k$ that is a quadratic twist of $E$ satisfies

$$\operatorname{rank} E'(k)\le R.$$

The source presents this as a direct consequence of Honda's conjecture, so it is not an independent assertion there. The supplied text gives no resolution status.