Shelah's chain-length conjecture for eventual domination

Let $\theta$ be an infinite successor cardinal and let $\mu\geq\theta$. A chain in $({}^{\theta^{++}}\mu,\ll_{\theta})$ is a sequence ordered by the relation $\ll_{\theta}$.

Shelah's chain-length conjecture. For every $\mu\geq\theta$, there are no chains in

$$({}^{\theta^{++}}\mu,\ll_{\theta})$$

of length $\mu^+$.

This conjecture extends Shelah's nonexistence results for long chains under eventual domination. Its status is not specified in the supplied text.