Question: Is it true that $[\binom kr(\frac rk)^r(1-\frac rk)^{k-r}]^k\le(\frac rk)^r(1-\frac rk)^{k-r}$ for all integers $0\le r\le k$? Is it further true that the inequality is strict for $0<r<k$ with $(r,k)\neq(1,2)$?
Background: Consider a matrix $X_{ij}$ of iid Bernoullis, $1\le i,j\le k$ with success probability $r/k$. I want to know if it is less likely to hit the expectation $r$ in each row, than to hit the expectation $r$ in the first row in a given order. In the context of my research, the $k$ in the exponent provides an upper bound for a critical value, which I don't need, but that I'd like to have.
Progress: I have verified the inequality for $r=2$ using calculus, and the proof does not generalize well. Further, I verified numerically that the inequality holds up to $k=100$.
Since the problem can be stated easily with only a few Bernoullis and this serves the purpose of establishing a simple upper bound, I'm hoping for an intuitive (maybe even short) proof. I tried to come up with a coupling, but failed so far. Induction may also be an option.
The equality clearly holds for $r = 0, k$. Let us consider the case $r \neq 0, k$ and $k \geq 3$. Consider the following bound on binomial coefficient: \begin{align*} \log \left( \binom{k}{r} \right) &\leq \frac{1}{2} \log\left( \frac{k}{2\pi(k - r)r} \right) + k \cdot h(r/k) \\ & \leq \frac{1}{2} \log\left( \frac{1}{4}\right) + k \cdot h(r/k) \\ & \leq -\log(2) + k \cdot h(r/k) \\ & \leq -h(r/k) + k \cdot h(r/k) \\ & \leq -(k -1) \left[ \frac{r}{k} \log\left( \frac{r}{k} \right) + \left(1 - \frac{r}{k} \right)\log\left(1 - \frac{r}{k} \right)\right], \end{align*} where $h(\alpha) = -\alpha \log(\alpha) + (1-\alpha)\log(1- \alpha)$ denotes the binary entropy function. On rearranging the above equation, we obtain the desired inequality. Note that in the second step, the inequality is actually strict implying that the overall inequality is also strict.