Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference $$ \left\vert P(X>c|X_1\leq a,X_2\leq a)-P(X>c)\right\vert $$ where $a$ and $c$ are constants in a neighborhood of $np$ and $2np$, respectively. Specifically, $a\in[np,np+o(n^{2/3})]$, and $c\in[2np,2a)$. These ranges will help control the rate of convergence. Assume that the conditioning set has probability 1 asymptotically.

The reason I'm interested in bounding the quantity above is because I need to approximate the conditional probability with the normal tail probability (e.g. by Berry-Esseen bound).

**Question**
Are there any references for this result? I've tried writing out the probabilities explicitly, but can't seem to get anywhere with the calculations.

**Edit**
By some elementary probability, I can show that on one hand
$$
P(X>c|X_1\leq a,X_2\leq a)=\frac{P(X>c,X_1\leq a,X_2\leq a)}{P(X_1\leq a,X_2\leq a)}\leq \frac{P(X>c)}{P(X_1\leq a,X_2\leq a)}\leq P(X>c)(1+o(1))
$$
where $o(1)$ is the rate as $n\to\infty$.
On the other hand,
\begin{align*}
P(X>c\mid X_1\leq a,X_2\leq a)&=\frac{P(X>c,X_1\leq a,X_2\leq a)}{P(X_1\leq a,X_2\leq a)}\\
&= \frac{P(X>c)}{P(X_1\leq a,X_2\leq a)}- \frac{P(X>c,\{X_1> a\}\cup\{X_2> a\})}{P(X_1\leq a,X_2\leq a)}\\
&\geq P(X>c)- \frac{P(X>c,\{X_1> a\}\cup\{X_2> a\})}{P(X_1\leq a,X_2\leq a)}\\
&\geq P(X>c)(1-o(1))
\end{align*}
where I claim that $P(X>c,\{X_1> a\}\cup\{X_2> a\})/P(X_1\leq a,X_2\leq a)=o(P(X>c))$.

This suggests the stronger result that $$ \lim_{n\to\infty}\frac{P(X>c\mid X_1\leq a,X_2\leq a)}{P(X>c)}=1 $$

Is the work above correct? Have I missed something or been careless about the convergence rates $1\pm o(1)$?