The gap and non-gap conjecture for minimal Terracini loci in projective space

For $n\ge 3$, let $\mathbb{T}(n,d;x)'$ denote the set of minimally Terracini sets of $x$ points in $\mathbb{P}^n$ with respect to $\mathcal{O}_{\mathbb{P}^n}(d)$: these are sets $S$ such that $h^0(\mathcal{I}_{2S}(d))>0$, $h^1(\mathcal{I}_{2S}(d))>0$, $\langle S\rangle=\mathbb{P}^n$, and $h^1(\mathcal{I}_{2A}(d))=0$ for every proper subset $A\subsetneq S$. The gap and non-gap conjecture. Fix an integer $c\ge 2$. Then there exists an integer $d_0(c,n)$ such that for all $d\ge d_0(c,n)$ there are integers $x_i$, $1\le i\le c$, and $y_i$, $1\le i\le c-1$, such that

$$x_1<y_1<x_2<y_2<\cdots <x_{c-1}<y_{c-1}<x_c,$$

$\mathbb{T}(n,d;x_i)'\ne\emptyset$ for all $i=1,\dots,c$, and $\mathbb{T}(n,d;y_i)'=\emptyset$ for all $i=1,\dots,c-1$. This predicts arbitrarily many alternating nonempty and empty cardinalities of minimal Terracini loci in higher-dimensional projective spaces, extending the phenomenon proved in the plane.