Gal sum for square free integer

I have a question regarding this paper by Tenenbaum and Bréteché. They define $$ S_\alpha(\mathcal{M}) = \sum_{m,n\in\mathcal{M}} \frac{(m,n)^\alpha}{[m,n]^\alpha} = \sum_{m,n\in\mathcal{M}} \biggl( \frac{(m,n)^2}{mn} \biggr)^\alpha \quad\text{and}\quad \Gamma_{\alpha}(N) = \sup_{\#\mathcal{M}=N} \frac{S_{\alpha}(\mathcal{M})}{N}, $$ where $\mathcal{M}$ is a finite set of $N$ integers. They also denote by $\mathbb{N}_1$ the set of square-free integers, and $\Gamma_{\alpha}^*(N)$ as the analogue of $\Gamma_{\alpha}(N)$ obtained by imposing $\mathcal{M} \subset \mathbb{N}_1$.

They restrict their study to the case $\alpha = \frac{1}{2}$, and denote $S(\mathcal{M}) = S_{1/2}(\mathcal{M})$, $\Gamma(N) = \Gamma_{1/2}(N)$, and $\Gamma^*(N) = \Gamma_{1/2}^*(N)$. They prove (see Theorem 1.1) that as $N \to \infty$, $$ \Gamma(N) = \mathcal{L}(N)^{2\sqrt{2}+o(1)}, $$ and remark (see eq. (1.5)) that their approach also yields $$ \Gamma^*(N) = \mathcal{L}(N)^{2+o(1)}, $$ where $$ \mathcal{L}(x) := \exp\biggl( \sqrt{\frac{\log x \, \log_3 x}{\log_2 x}} \biggr). $$

I have two questions:

1. I do not understand how the exponent changes from $2\sqrt{2}$ to $2$ in the square-free case. Why does the restriction to square-free integers reduce the exponent by a factor of $\sqrt{2}$?

2. If we use the sets $\mathcal{M}$ from paper 2, can we prove $$ \Gamma^*(N) = \mathcal{L}(N)^{2+o(1)}? $$

Thank you in advance for your help.