Asymptotic binomial-sum conjecture for even moments

Let $k\geq 1$, and let $j$ tend to infinity. The two binomial sums below involve the even moments of the centered binomial distributions of sizes $2j$ and $2j-1$.

Even-moment asymptotic conjecture. There is a constant $c(k)>0$ such that, as $j\to\infty$,

$$2^{-2j+1}\sum_{n=1}^j(2n)^{2k}\binom{2j}{j+n}\sim c(k)j^k, \qquad 2^{2j}\sum_{n=1}^j(2n-1)^{2k}\binom{2j-1}{j+n-1}\sim c(k)j^k.$$

The conjecture is used to obtain the higher-derivative representation conjecture. The authors report verification by computer algebra only for $k\in\{3,\dots,15\}$, so the general asymptotic remains open.