hat{S}}}$ denotes the vector of LOCO parameters arising from the bootstrap distribution corresponding to the empirical measure associated to the $n$ triplets $\left\{  (X_i,Y_i,\xi_i), i \in
    \mathcal{I}_{2,n} \right\}$.

Next, $$\mathbb{P}\left( \sqrt{n} \| \gamma_{{\widehat{S}}} - \hat{\gamma}_{{\widehat{S}}}  \|_\infty
    \leq \hat{t}^*_{\alpha} \right) \geq  
    \mathbb{P}\Big( \sqrt{n} \| \hat{\gamma}^*_{{\widehat{S}}} - \hat{\gamma}_{{\widehat{S}}}   \|_\infty \leq \hat{t}^*_
    \alpha  \left | (X_i,Y_i,\xi_i), i \in
    \mathcal{I}_{2,n} \right. \Big) 
    -( A_1 + A_2 + A_3),$$ where $$\begin{aligned}
    A_1 & = \sup_{t >0} \left| \mathbb{P}\left( \sqrt{n} \| \hat{\gamma}_{{\widehat{S}}} - \gamma_{{\widehat{S}}}   \|_\infty
    \leq t \right) - \mathbb{P}( \| Z \|_\infty \leq t ) \right|,\\
    A_2 & = \sup_{t>0} \left|  \mathbb{P}( \| Z \|_\infty \leq t ) - \mathbb{P}( \|
    \hat{Z} \|_\infty \leq t )      \right|,\\
    \text{and} & \\
    A_3 & = \sup_{t >0} \Big|
   \mathbb{P}\Big( \sqrt{n} \| \hat{\gamma}^*_{{\widehat{S}}} - \hat{\gamma}_{{\widehat{S}}}
   \|_\infty \leq t   \left | (X_i,Y_i,\xi_i), i \in
    \mathcal{I}_{2,n} \right. \Big) -  \mathbb{P}( \| \hat{Z} \|_\infty \leq
    t ) \Big|,\end{aligned}$$ with $Z = \Sigma_{{\widehat{S}}}^{1/2} Z_1$ and $\hat{Z} = \widehat{\Sigma}_{{\widehat{S}}} Z_2$.

Then, $A_1 \leq C \mathrm{E}_{1,n}$ by and $A_2 \leq
C \mathbb{E}_{2,n} + \frac{1}{n} $, by applying the Gaussian comparison Theorem \[thm:comparisons\] on the event $\mathcal{E}_n$ that holds, whereby $\mathbb{P}(\mathcal{E}_n^c) \leq \frac{1}{n}$ as argued in the proof of . Finally the bound on $A_3$ follows from applying Theorem 2.1 in [@cherno2] to the bootstrap measure, conditionally on $(X_i,Y_i,\xi_i), i \in
    \mathcal{I}_{2,n}$, just like it was done in the proof of . In this case, we need to restrict to the even $\mathcal{E}_n$ to ensure that the minimal variance for the bootstrap measure is bounded away from zero. To that end, it will be enough to take $n$ large enough so that 