Coinduction preservation conjecture for compatible indexing categories

Let $(\mathcal O_a,\mathcal O_m)$ be a compatible pair, and let $H/K$ be admissible for $\mathcal O_m$. Write $\operatorname{Map}^K(H,-)$ for coinduction from $K$ to $H$.

Coinduction preservation conjecture. Coinduction restricts to a functor

$$\operatorname{Map}^K(H,-)\colon i_K^*\mathcal O_a\to i_H^*\mathcal O_a.$$

In this case, coinduction is said to preserve $\mathcal O_a$.

This conjecture asserts closure of the relevant indexing-category maps under coinduction and is the condition needed for constructing norm functors by left Kan extension. The source gives no resolution of the conjecture.