The internal-left-adjoint compact-generation conjecture for linear categories

Let $R$ be a commutative ring spectrum, and let $C$ be a dualizable $R$-linear presentable stable $\infty$-category. The coevaluation morphism is the morphism

$$\mathbf{Mod}_R\to C\otimes_R C^\vee.$$

Internal-left-adjoint compact-generation conjecture. If the coevaluation morphism is an internal left adjoint, then $C$ is compactly generated. The paper states that no counterexamples are known; related results establish the conclusion over a discrete field and slightly more general bases, but the general case remains open.