The ring compatibility conjecture for the modified Hochschild-Kostant-Rosenberg isomorphism

Let $I$ be the Hochschild-Kostant-Rosenberg isomorphism and let $I'$ be its modification obtained by composing it with wedging by $\operatorname{td}_X^{1/2}$. For a homogeneous element $v$ in ${\mathrm{Hom}}_X^*(\bigoplus_i \Omega_X^i[i],\mathscr O_X)$, let $e(v)$ be the upper-triangular matrix defined by the product formula $v*v'=p(e(v)\circ e(v'))$, and let $e'(v)$ be the matrix obtained by following the arrows around the preceding diagram after replacing $I$ by $I'$. Ring compatibility conjecture. Replacing $I$ by $I'$ in the preceding analysis, and $I^{\mathrm{HKR}}$ by $I^K$, yields $e=e'$. This conjecture asserts that the Todd-class modification repairs the failure of the ordinary Hochschild-Kostant-Rosenberg map to respect the relevant ring structure; the source gives no resolution of the claim.