Computed deviations from Fröberg's conjectured Hilbert series

For generic linear forms $l_i$, let

$$Q_{n,r,d}(t)=H_{k[x_1,\ldots,x_n]/(l_1^d,\ldots,l_r^d)}(t),$$

and let

$$F_{n,r,d}(t)=\left[\frac{(1-t^d)^r}{(1-t)^n}\right].$$

Here $F_{n,r,d}(t)$ is the Hilbert series conjectured by Fröberg for generic degree-$d$ forms in the cases under consideration. The computed deviation claim. The following equalities hold:

$$Q_{12,14,2}(t)-F_{12,14,2}(t)=64t^6,$$ $$Q_{13,15,2}(t)-F_{13,15,2}(t)=13t^6+t^7,$$ $$Q_{13,16,2}(t)-F_{13,16,2}(t)=t^6,$$ $$Q_{9,11,3}(t)-F_{9,11,3}(t)=t^8+154t^9+t^{10},$$ $$Q_{9,12,3}(t)-F_{9,12,3}(t)=12t^8.$$

These are computational claims rather than theorems: the source notes that the calculations used random linear forms and that the observed inequality could theoretically result from an atypical choice, although this is considered unlikely. Their status as general mathematical assertions therefore remains open.