The separation of increasing and decreasing Turing-degree chains

For $p\in\mathbb{N}^{\mathbb{N}}$, let $\mathsf{TDINC}(p)$ consist of sequences $\langle p_0,p_1,p_2,\ldots\rangle$ such that, for every $i$, $p<_{\mathrm{T}}p_i<_{\mathrm{T}}p_{i+1}$, and let $\mathsf{TDDEC}(p)$ consist of sequences such that, for every $i$, $p<_{\mathrm{T}}p_{i+1}<_{\mathrm{T}}p_i$. The chain-separation conjecture.

$$\mathsf{TDINC}\mathop{\not\leq_{\mathrm{W}}}\mathsf{TDDEC}.$$

This is presented as the next conjecture implying the strict separation between $\mathsf{NON}^\omega$ and $\mathsf{NON}^\infty$; its resolution is left open in the supplied text.