Antieau–Gepner–Heller theorem of the heart conjecture for bounded t-structures

Let $\sC$ be a small stable $\infty$-category equipped with a bounded t-structure, and let $\sC^\heartsuit$ be its heart. Let $\sD^b(\sC^\heartsuit)$ be the bounded derived category of the heart, with the canonical map

$$\K(\sD^b(\sC^\heartsuit))\to\K(\sC).$$

Antieau–Gepner–Heller's theorem of the heart conjecture. The map

$$\K(\sD^b(\sC^\heartsuit))\to\K(\sC)$$

is an equivalence of spectra. This is the nonconnective theorem-of-the-heart assertion for bounded t-structures; the source gives no resolution in the provided text.