Claim: Experimental data seems to suggest that
$$ {n \choose 1^a b}{n \choose 2^a b}{n \choose 3^a b}\cdots {n \choose m^a b} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{ab+3b}\bigg) $$ where $a$ and $b$ are a fixed positive integers and $m$ is the largest positive integer such that $m^a b \le n$.
Note that the above asymptotic are supported by the data even if we relax the condition that $a,b$ are integers and allow them to be reals $a > 0, b > 0$ and replace $k^a b$ with $\lfloor k^a b\rfloor$.
As an illustration, for $a = 3, b = 1$, the $\%$ error between the asymptotic and the actual product is shown below.
Note: Posted in MO since in it unanswered in MSE
Update 19-Dec-19: Combined the two individual claims into a single claim based on experimental data.
