Veronese-surface Betti-table conjecture

Let $X_{d\Sigma}$ be the $d$-fold Veronese surface, with $\Sigma=\operatorname{conv}\{(0,0),(1,0),(0,1)\}$, and let $b_i$ and $c_i$ denote the entries on the linear and quadratic strands of its graded Betti table. Write

$$N_{(d\Sigma)^{(1)}}=\left|(d\Sigma)^{(1)}\cap\mathbb{Z}^2\right|=\frac{(d-1)(d-2)}{2}.$$

Veronese-surface Betti-table conjecture. If $d\geq2$, then the last non-zero entry on the linear strand is

$$b_{d(d+1)/2}=\frac{d^3(d^2-1)}{8},$$

while if $d\geq3$, the first non-zero entry on the quadratic strand is

$$c_g=\binom{N_{(d\Sigma)^{(1)}}+8}{9}.$$

These formulas make precise the expected pattern in the graded Betti tables of Veronese surfaces; the source presents them as an extrapolation from computed data, and no resolution is supplied here.