The compatibility of the modified HKR maps with Hochschild and harmonic structures

Let $X$ be a compact smooth space. The modified Hochschild-Kostant-Rosenberg maps $I^K$ and $I_K$ are the maps obtained by twisting the Hochschild-Kostant-Rosenberg isomorphism by $\bigwedge\operatorname{td}_X^{1/2}$, where $\operatorname{td}_X^{1/2}$ is the square root of the Todd class. Compatibility conjecture. The maps $(I^K,I_K)$ form an isomorphism between the Hochschild and harmonic structures of $X$. This would establish that the modified Hochschild-Kostant-Rosenberg isomorphism gives the correct compatibility between the Hochschild and harmonic structures, consistently with the Mukai vector and its functorial properties.