**UPDATE (2022-07-13)**. The generating function for $A_k$ can be expressed as
$$\sum_{k\geq0} A_k t^k = {\cal L}_{x_1,\dots,x_n,y_1,\dots,y_n} \sum_{\lambda} e_{\lambda}(x_1,\dots,x_n)\cdot m_{\bar\lambda}(y_1,\dots,y_n)\cdot t^{\mathrm{sum}(\lambda)},$$
where summation is done over all partitions $\lambda$ whose Young tableau fits the $n\times n$ square; $\bar\lambda$ is the partition whose Young tableau complements that of $\lambda$ in the $n\times n$ square; $e$ and $m$ are elementary and monomial symmetric polynomials respectively; and $\cal L$ is the Laplace transform evaluated at $1$, which replaces each $x_i^d$ or $y_j^d$ with $d!$.

With this formula I was able to extend computations and confirm the conjecture for $n\leq 10$. The data is uploaded to OEIS A261602 and OEIS A261603.

Below is my original answer presenting data for $n\leq 6$.

I've computed values of $|A_k|$ for $0\leq k\leq \lfloor n^2/2\rfloor$ and $n\leq 6$:

$n=1:$ 1

$n=2:$ 4, 8, 10

$n=3:$ 216, 648, 1188, 1668, 1944

$n=4:$ 331776, 1327104, 3151872, 5695488, 8608896, 11446272, 13791744, 15326208, 15858432

$n=5:$ 24883200000, 124416000000, 360806400000, 787138560000, 1426595328000, 2262299258880, 3240594432000, 4283587584000, 5304730521600, 6222411878400, 6968709089280, 7493189990400, 7763310604800

$n=6:$ 139314069504000000, 835884417024000000, 2855938424832000000, 7259810955264000000, 15220062093312000000, 27765294052147200000, 45532546213478400000, 68600569724928000000, 96440964380098560000, 127985462154362880000, 161777817980986982400, 196164002436769382400, 229476155622594969600, 260178812386069708800, 286962944406552576000, 308788668410898677760, 324887962565624463360, 334743605500457779200, 338060641751949312000

So the conjecture is confirmed numerically for $n\leq 6$.

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