Coordinate convergence conjecture for fixed-point-free polydisc self-maps

Let $N\ge1$ and let $F:\mathbb D^N\to\mathbb D^N$ be holomorphic and without fixed points. Write $F^{\circ n}$ for the $n$-fold iterate and let $\pi_j(z)=z_j$ be projection onto the $j$-th coordinate. Polydisc coordinate-convergence conjecture. There exist $j\in\{1,\ldots,N\}$ and $\sigma\in\partial\mathbb D$ such that

$$\lim_{n\to\infty}\pi_j(F^{\circ n}(z))=\sigma$$

for every $z\in\mathbb D^N$. This would rule out the more complicated target-set behavior suggested by general inclusion results for fixed-point-free self-maps of polydiscs; the source states it as generally believed but unresolved.