The freeness conjecture for the Flenner–Zaidenberg first series

Let $C_{d,a}\to\text{ }$ denote a curve in the family ${\mathcal F}{\mathcal Z}_1(d,a)$, where the family consists of the rational cuspidal curves described in the source, and let $mdr(f)$ be the minimal degree of a Jacobian syzygy of its defining equation $f$.

Freeness conjecture for ${\mathcal F}{\mathcal Z}_1$. The curves $C_{d,a}\in {\mathcal F}{\mathcal Z}_1(d,a)$ are free divisors and $mdr(f)=2$ for all possible values of $(d,a)$ when $d\geq 5$. In particular,

$$\tau(C_{d,a})=d^2-4d+7$$

for every such pair $(d,a)$.

The claim concerns the first of the three infinite families of rational cuspidal plane curves with at least three cusps. No verification or resolution is stated in the supplied context.