There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices: $$ P_n \, := \, \frac{1}{C_n} \, \sum_{A \, \in \, \text{SYT}(n,n)} A $$ Note that $P_n=(p_{ij})$ is a rational matrix with entries monotone increasing in rows and columns. Also, by the 180$^\circ$ rotational symmetry, $p_{ij} + p_{3-i,n+1-j} = 2n+1$. For example, $p_{11}=1$, $p_{2n}=2n$.

The asymptotic density of $P_n$ as $n\to \infty$ is easy to obtain by a direct calculation or via the Brownian excursion (see e.g. here or there), but my question is different.

**Question.** Let $\beta(n):= \min_{(ij)\ne (kl)} |p_{ij}-p_{kl}|$. Is it true that $\beta(n) = o(1)$?

It would be cool if there was an easy way to see this. I really want a generalization of this result to all large partitions, but at the moment even this is confounding.

UPDATE (May 18, 2019):

Let me explain the motivation behind the question. Recall the 1/3-2/3 conjecture that every poset $\mathcal P=(X,\prec)$ that is not a linear order contains two elements $x,y\in X$ such that
$$\frac13 \le P(x\prec y) \le \frac23
$$
For width 2 posets this was shown by Linial in this paper, but I thought that for shapes $(n,n)$ one can improve $1/3$ to perhaps $(1/2-\varepsilon)$, since we know so much about Catalan numbers (including the average of Catalan objects). Linial's proof cannot be easily improved, unfortunately. Now, a beautiful Kahn-Linial proof of the weaker $1/2e$ bound starts with the average LE of $\mathcal P$. If $\beta(n)=o(1)$, their argument plus the (earlier) Grünbaum Theorem implies the $(1/e-\varepsilon)$ bound, already a nice result.

Now, Richard's calculaitons give $p_{17} \to 9949/1024 \approx 9.7158$, $p_{23} \to 9.75$. This means that taking $x=(1,7)$ and $y=(2,3)$ in $(n,n)$ gives $$0.3553 < \frac1e \left(1-\frac{35}{1024}\right) < P(x\prec y) < \frac2e \left(1+\frac{35}{1024}\right) < 0.6447 $$ for $n$ large enough (unless I miscalculated). This improvement over $1/3$ bound is good to know, but surely one can do better.