Freeness criterion for members of a pencil containing a unique non-reduced curve

Let $hh_1^{r_1}\cdots h_s^{r_s}=0$ be the equation of the unique non-reduced curve, where ${h=0}$ is reduced, $\operatorname{deg}(h_i)=m_i\geq 1$, and $r_i\geq 2$. Let $D_k$ be a member of the pencil, and let $D^{\mathrm{sg}}$ denote the singular locus scheme of the pencil. The reduced structure of $D_k$ has its Jacobian ideal defined by the partial derivatives of a reduced equation for $D_k$. Freeness criterion. $D_k$ is free with exponents

$$\left(2n-2-\sum_{i=1}^{s}(r_i-1)m_i,\,n(k-2)+1\right)$$

if and only if $D_k\supseteq D^{\mathrm{sg}}$ and the Jacobian ideal of the reduced structure of $D_k$ is locally a complete intersection.

This extends the preceding freeness criteria to the case where the pencil contains a non-reduced curve, by removing the divisor contributed by its repeated components. The statement is presented in the source as something the authors believe to be true, and no resolution is supplied here.