Two special cases of the minimum distance conjecture for complete intersections

Let $\mathbb{X}$ be a finite set of reduced points in $\mathbb{P}^{s-1}$, and suppose that $I=I(\mathbb{X})$ is a complete intersection generated by $f_1,\ldots,f_c$, where $c=s-1$, $d_i=\deg(f_i)$ for $i=1,\ldots,c$, and $2\leq d_i\leq d_{i+1}$ for all $i$. Two special cases of the minimum distance conjecture. (a) One has

$$\delta_I(1)\geq (d_1-1)d_2\cdots d_c.$$

(b) If $f_1,\ldots,f_c$ are quadratic forms, then

$$\delta_I(d)\geq 2^{c-d}\qquad\text{for }1\leq d\leq c,$$

or, equivalently,

$$\operatorname{hyp}_I(d)\leq 2^c-2^{c-d}\qquad\text{for }1\leq d\leq c.$$

These are explicitly identified as two special cases that remain open. The first is attributed to Tohăneanu and Vântuyl; the second concerns complete intersections generated by quadrics.