Fiori's conjecture on large and small outliers for the least prime

Let $\epsilon>0$, and let $E_\epsilon(x,y)$ denote the counting function introduced in the paper for the relevant large outliers of $P(n)$; the supplied context does not reproduce its definition. Let $P(n)$ and $\phi(n)$ be as above, and let $\ll$ denote an upper bound up to a constant. Fiori's outlier conjecture. The paper conjectures the following four assertions: for $\epsilon>1$ as $y\to\infty$,

$$\frac{y^{1-\epsilon}-x^{1-\epsilon}}{(1-\epsilon)\log(y)}\ll |E_\epsilon(x,y)|\ll \frac{\log(x)^2x^{1-\epsilon}}{\epsilon-1};$$

for $\epsilon=1$ as $y\to\infty$,

$$\log\log(y)\ll |E_\epsilon(x,y)|\ll \log(y)^3-\log(x)^3;$$

for $\epsilon>1$ as $y\to\infty$,

$$\frac{y^{1-\epsilon}-x^{1-\epsilon}}{(1-\epsilon)\log(y)}\ll |E_\epsilon(x,y)|\ll \frac{\log(y)^2y^{1-\epsilon}-\log(x)^2x^{1-\epsilon}}{1-\epsilon};$$

and

$$\limsup_{n\to\infty}\frac{P(n)}{\phi(n)\log(n)\log(\phi(n))}=2.$$

In particular, the source concludes that $E_\epsilon$ is finite for all $\epsilon>1$ and infinite for all $\epsilon\leq1$. The supplied statement itself repeats $\epsilon>1$ in its first and third cases, so that apparent inconsistency is preserved rather than corrected.