Maximizing the sum of adjacent pairs of elements

Question

I encountered the following interesting problem on stackoverflow:

Given numbers $a(1)<\cdots<a(n)$, find a permutation $\pi$ that maximizes $$\sum_{i=1}^{n-1} a(\pi(i)) a(\pi(i+1)).$$

The answers there claim that the answer doesn't depend on the numbers themselves, and is always given by the order $$\ldots,a(n-5),a(n-3),a(n-1),a(n),a(n-2),a(n-4),\ldots$$ However, the proof (given in the comments) isn't convincing.

Is the answer on stackoverflow correct? Can we prove it?