Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.

I know of the following:

$P$ is positive definite and elementwise nonnegative. Moreover, $p_{jk}p_{ii} \ge p_{ji}p_{ik}$ for any $i,j,k$.

I can verify that the statement is true for $n=2$, but I don't know how to work with $n$ large. Playing around with randomly generated matrices in Matlab seems to suggest that the statement is true. Any hint or suggestion would be greatly appreciated.

I've googled out that a very similar statement was put as a conjecture in this paper: *Optimization of an on-chip active cooling system based on thin-film thermoelectric coolers* (http://dl.acm.org/citation.cfm?id=1870955)

**Edit:** Perhaps someone can solve this easier question: Is there a positive semi-definite and elementwise nonnegative $P$ and diagonal $V$ such that $PVPVP$ is not elementwise nonnegative?