Kivinen's conjectural description of Hilbert-scheme homology for rational-component curves

Let $C$ be the unique compactification with rational components and no other singularities of the curve $\{x^n=y^{dn}\}$. Let

$$V:=\bigoplus_{m\geq 0} H_*(C^{[m]},\mathbb{Q}).$$

The algebra $A_n$ is the algebra acting on this homology described in the preceding results. Kivinen's conjecture. As a bigraded $A_n$-module, one has

$$V\cong \frac{\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]}{\sum_{i\neq j}\sum_{k=1}^{d}(x_i-x_j)^k\ker(\partial_{y_i}-\partial_{y_j})^k}.$$

This conjecture is motivated by computations for reducible planar curves and by the analogous description of ordinary equivariant homology of affine Springer fibers. The source presents it as conjectural and gives no resolution.