The heart-equivalence conjecture for K-theory of bounded t-structures

Let $E$ be a small stable $\infty$-category equipped with a bounded $t$-structure, and let $E^\heartsuit$ denote its heart, an abelian category. There is a natural map of nonconnective $K$-theory spectra

$$\mathrm{K}(E^\heartsuit)\longrightarrow\mathrm{K}(E).$$

Heart-equivalence conjecture. This natural map should be an equivalence of nonconnective $K$-theory spectra. The assertion would identify the $K$-theory of a boundedly $t$-structured stable category with that of its heart and implies the corresponding negative $K$-theory vanishing; it remains open in the stated generality.