Douglas's D-geometry conjecture for D-brane moduli

Let $Y$ be a scheme over $\mathbb{C}$, let $r\in\mathbb{N}$, and let $\mathfrak M^{{0^{A\!z}}^f}_r(Y)$ be the moduli stack of rank-$r$ D0-branes on $Y$. Write

$$Y^{nc}_r:=\left(\mathfrak M^{{0^{A\!z}}^f}_r(Y),\mathcal O_{\mathfrak M^{{0^{A\!z}}^f}_r(Y)},\mathcal E_{\mathfrak M^{{0^{A\!z}}^f}_r(Y)}\right)$$

for this Azumaya stack with its fundamental module, and let $\mathcal X_r$ denote Douglas's configuration space of $r$ D-branes. Douglas's D-geometry conjecture. An atlas for $Y^{nc}_r$ corresponds to $\mathcal X_r$. If $Y$ is a Kähler manifold, the irreducible component of $Y^{nc}_r$ containing all length-$r$ zero-dimensional $\mathcal O_Y$-modules supported at $r$ distinct points of $Y$ carries an associated formal Kähler geometry. This geometry can satisfy the mass conditions of Douglas and Douglas–Katz–Ooguri if and only if $Y$ is Ricci flat. The conjecture proposes a geometric identification of the D-brane moduli stack with Douglas's D-geometry; the Ricci-flat condition reflects the expected mass constraints, while the general validity of the construction is not established.