Shelah's conjecture on increasing chains modulo small-set ideals

Let $\theta$ and $\mu$ be cardinals with $\mu>\theta$. Consider the partial order of functions ${}^{(\theta^{+3})}\mu$ ordered modulo the ideal $[\theta^{+3}]^{\leq\theta}$ of subsets of $\theta^{+3}$ of cardinality at most $\theta$. Shelah's conjecture. There is no increasing sequence of length $\mu^+$ in ${}^{(\theta^{+3})}\mu$ modulo $[\theta^{+3}]^{\leq\theta}$. This would generalize the paper's result ruling out an $\omega_4$-long sequence in ${}^{\omega_3}\omega_3$ modulo the ideal of countable sets; the general assertion is presented as a hoped-for result and remains open in the supplied text.