Hypergeometric formulas for ordered multi-degree contributions

Let $\boldsymbol{d}=(d_0,d_1,d_2)$ be an ordered multi-degree, and let $\operatorname{C}_{\operatorname{ord}}(\boldsymbol{d})$ denote its ordered multi-degree contribution. For positive integers $d$, $d_1$, and $d_2$, the relevant binomial coefficients are defined in the usual way.

Hypergeometric contribution conjecture. The ordered multi-degree contributions satisfy

$$\operatorname{C}_{\operatorname{ord}}(d,0,0)=\dfrac{1}{d^2}{4d-1\choose d}\qquad (d\geq 1),$$

and

$$\operatorname{C}_{\operatorname{ord}}(d_1,d_2,0)=\dfrac{6}{d_1d_2}{4d_1+2d_2-1\choose d_1-1}{4d_2+2d_1-1\choose d_2-1}\qquad (d_1,d_2\geq 1).$$

The first formula is supported by its relationship, after taking a logarithm, with the diagonal term of the $3$-Kronecker quiver, whose corresponding conjecture of Gross was proved by Reineke. The second formula is not immediately explained by the same heuristic, so the general formulas remain a conjectural description of these contributions.