Norm functors for compatible indexing categories

Let $(\mathcal O_a,\mathcal O_m)$ be a compatible pair of indexing categories, and let $H/K$ be an $\mathcal O_m$-admissible $H$-set.

Norm functor conjecture. There is a norm functor

$$N_K^H\colon i_K^*\mathcal O_a\text{-}\operatorname{Mackey}\to i_H^*\mathcal O_a\text{-}\operatorname{Mackey}$$

that is symmetric monoidal with respect to the box product.

Such norm functors are expected to provide the additively incomplete analogue of the norm structure on Mackey functors and to assemble into the external form of bi-incomplete Tambara functors. The source gives no resolution of this conjecture.