How to prove the following floor function inequality?

Question

For any $M\in\mathbb{Z}^+$ and $n\ge\lceil\log_2(M) \rceil, n\in\mathbb{Z}$, we have $$ M^2\log_2(M)\le 4\sum_{i=1}^n\left(M-\sum_{k=0}^{2^{i-1}-1}\left\lfloor\frac{M+k}{2^i} \right\rfloor\right)\left(\sum_{k=0}^{2^{i-1}-1}\left\lfloor\frac{M+k}{2^i} \right\rfloor\right) < M^2\left[\log_2(M)+\frac12\right] $$ where the equality holds if and only if $\log_2(M)\in\mathbb{Z}$.