Cone conjecture for characteristic-dependent Betti numbers of graph ideals

Let $G$ be a connected graph on $n$ vertices, and let $I(G)\subseteq k[x_1,\dots,x_n]$ be its edge ideal. Introduce a new vertex $n+1$ and set

$$J=I(G)+x_{n+1}(x_1,\dots,x_n)\subseteq k[x_1,\dots,x_n,x_{n+1}],$$

which is the edge ideal of the cone over $G$. Betti numbers are computed over the coefficient field $k$.

Cone conjecture. If the Betti numbers of $I(G)$ depend on the field, then the Betti numbers of $J^2$ also depend on the field.

The conjecture is motivated by numerous computer experiments and asks whether adjoining a cone vertex preserves characteristic dependence at the level of the square. Its general validity is not established in the source.