The sound reflection hierarchy classification conjecture

For $n\in\mathbb{N}$, let $0^{(n)}$ denote the $n$th Turing jump, let $B\Sigma_1$ be the stated base theory, and let $\Sigma_{n+1}$-soundness have its usual meaning.

Sound reflection hierarchy classification conjecture. For every $n\in\mathbb{N}$, both of the following hold: (1) there exists a $0^{(n)}$-recursive sequence $(T_n)_{n<\omega}$ of $\Sigma_{n+1}$-sound recursively enumerable extensions of $B\Sigma_1$ such that, for each $n$, $T_n$ proves the $\Sigma_{n+1}$-soundness of $T_{n+1}$; and (2) there is no $0^{(n)}$-recursive sequence $T(T_n)_{n<\omega}$ of $\Sigma_{n+1}$-sound recursively enumerable extensions of $B\Sigma_1$ such that, for each $n$, $T_n$ proves the $\Sigma_{n+1}$-soundness of $T_{n+1}$.

The source presents this as a hoped-for classification associated with a question about descending sequences in reflection hierarchies. The parser marks it open, and the source gives no resolution.