Eleven Swaps Not Thirty Four

Take a perfect binary min-heap holding exactly 15 elements, so there are 8 leaves, 4 nodes one level up, 2 above those, and the root.

Building it bottom up runs sift-down at every internal node. A node at height $h$ can sink at most $h$ levels, and a perfect heap of $n$ nodes has $\frac{n+1}{2^{h+1}}$ nodes at height $h$:

$\text{swaps} \le \sum_{h=1}^{3} \frac{15+1}{2^{h+1}} \cdot h$

Evaluate that worst case.