Convergence and complexity of the magnitude-matrix balancing algorithm

Let $M\in\mathbb{R}^{n\times n}$ be nonnegative, let $\theta\in(0,1)$, and let $T$ be the iteration horizon. Define the paper's simultaneous-coordinate heuristic Algorithm~ using the iterates $d_k[t]$. For a fixed tolerance $\epsilon$, consider the number of iterations needed to reach that tolerance.

Balancing-algorithm conjecture. Algorithm~ always converges. Moreover, the number of iterations required to reach a given tolerance is $O(\log(n))$ for fixed $\epsilon$, and $O(\sqrt{\epsilon^{-1}})$ for fixed $n$.

The conjecture is based on numerical experiments and is intended to justify a distributed heuristic for computing the upper-bound optimization. The source does not provide a convergence proof or establish the stated iteration-rate estimates.