Thick tensor-ideal exhaustion conjecture for dualizable spectra in the \Sp_{k,n}-local category

Let $0 \le i \le n+1$. For $i \le n$, let $\mathcal D_i$ be the full subcategory of dualizable objects $X$ in $\Sp_{k,n}^{\dual}$ that are retracts of $Y \wedge Z$ for some dualizable $Y$ and some finite spectrum $Z$ of type at least $i$, and set $\mathcal D_{n+1}=(0)$. A thick tensor-ideal exhaustion conjecture asserts that every thick tensor-ideal $\mathcal C$ of $\Sp_{k,n}^{\dual}$ is one of these subcategories:

$$\mathcal C=\mathcal D_i$$

Equivalently, the Balmer spectrum is

$$\Spc(\Sp_{k,n}^{\dual})=\{\mathcal D_1,\ldots,\mathcal D_{n+1}\},$$

with closure operator

$$\overline{\{\mathcal D_i\}}=\{\mathcal D_j\mid j\ge i\}.$$

This generalizes the Hovey--Strickland conjecture from the $K(n)$-local case $k=n$; the paper notes that the case $k=n$ has been investigated in detail, while the general case remains conjectural.