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Not quite what you asked, but too long for a comment. In a book in 1978 Lang conjectured that on a (quasi)minimal Weierstrass equation, we have $$\bigl|E(\mathbb{Z})\bigr|\le{C}^{\operatorname{rank}E(\mathbb{Q})},$$ where $C$ is an absolute constant. And assuming "standard conjectures", we have $$\operatorname{rank}E(\mathbb{Q})\ll\log{N_E}/\log\log{N_E}.$$ Since the conductor is smaller than the discriminant, combining these gives the conjecture that you quote, in slightly stronger form that one can take $\epsilon=c/\log\log\Delta$ for an absolute constant $c$.