This is a variant on Sneaky Recursive Non-Well-Orders where it was asked

Is there a recursive function $f$ such that whenever $a\in\mathcal{O}$, $f(a)$ is a Turing index for a linear non-well-order with no $H_a$ -computable descending chain?

The answer to the original question gave a single non-well-order with no hyperarithmetic descending chain at all. Instead can $f(a)$ be a non-well-order with an $H_a$-computable descending chain but no $H_b$-computable descending chain for $b <_{\mathcal{O}} a$ ?