Syzygy resolution conjecture for a specified plane curve

Let $R=k[x,y,z]$ and let $f=(x^2-y^2)z^{d-1} -(x^{d-1}-y^{d-1})x^2 -y^{d+1}$, with $d\geq 2$, and let $J_f$ be its gradient ideal. Syzygy-resolution conjecture. The maximal degree of a minimal generator of $\operatorname{Syz}(J_f)$ is $2d-2$. Moreover, the minimal graded free resolution of $J_f$ is

$$0 \longrightarrow R(-(3d-1))^2 \longrightarrow R(-2d)^3\oplus R(-(3d-2))\longrightarrow R(-d)^3 \longrightarrow R,$$

where the generating syzygies in degree $2d$ are the Koszul syzygies; in particular, the regularity of $R/J_f$ is $3d$. These assertions propose a precise description of the extremal syzygy degree and graded resolution for this example; the supplied text gives no resolution status.