Hausdorff dimension of level sets in Engel continued fraction

Let $[[b_1(x),\dots b_n(x),\dots]]$ be the Engel continued fraction expansion of $x\in(0,1)$. This paper is concerned with the growth of the partial quotients $b_n(x)$. We obtain the Hausdorff dimension of the sets $$ E_\phi=\left\{x\in(0,1):\lim_{n\to\infty}\frac{\log b_n(x)}{\phi(n)}=1\right\}, $$ for any non-decreasing $\phi$ satisfying $\lim\limits_{n\to\infty}(\phi(n + 1) - \phi(n)) = \infty$ and $\lim\limits_{n\to\infty}\phi(n + 1)/\phi(n) = 1$.