Recursive Fröberg conjecture for adjoining a generic form

Let $S=\Bbbk[x_1,\ldots,x_n]$, let $J\subset S$ be generated by generic forms, and let $f\in S$ be a generic form of degree $e$. Recursive Fröberg conjecture. For every integer $j$,

$$\dim_\Bbbk [S/(J,f)]_j=\max\{0,\dim_\Bbbk[S/J]_j-\dim_\Bbbk[S/J]_{j-e}\}.$$

This is the recursive Hilbert-function formulation of Fröberg's conjecture; it is used to predict the dimensions needed in the secant-variety application and remains open in general.