Dense hard-sequence conjecture for true unprovable coTHEOREMS

Let $H'$ be the set comprising $H$ and other dense sets of true sentences unprovable in $\mathcal{T}$, including the specified sets associated with universal Turing machines and Turing jumps. For a nondeterministic Turing machine $M$ accepting $\texttt{coTHEOREMS}$, let $\phi_i$ denote the $i$th arithmetic sentence in the relevant enumeration. Dense hard-sequence conjecture. There is a dense subset $H'_M$ of $H'$ such that $i\in H'_M$ if and only if $(\langle\phi_i,1^t\rangle)_{t\in\mathbb{N}}$ is a hard sequence for $M$. The conjecture identifies dense hard sequences arising from true unprovable sentences and would imply that $\texttt{coTHEOREMS}$ has dense hard sequences; the source notes that $H'$ and the associated notion of balance may not be fully well defined.