Pasten's conjecture on the Tamagawa product of elliptic curves

Let $E$) be an elliptic curve over $\mathbb{Q}$, with conductor $N=N(E)$ and Tamagawa product $\tau=\tau(E)$. Here $\ll$ denotes an absolute implied constant, and $\epsilon>0$ is arbitrary. Pasten's conjecture. For all elliptic curves

$$\tau \ll N^{\left(\frac{7}{3}\log 3+\epsilon\right)/\log\log N}.$$

The conjecture gives a conditional-looking refinement of bounds relating the Tamagawa product to the conductor, motivated by the $abc$ Conjecture and by the search for elliptic curves with large Tate–Shafarevich groups. Its status is not resolved in the supplied source.