General infinitesimal Terracini Lemma

Let $X\subseteq {\mathbb P}^r$ be an irreducible, non-degenerate, projective variety of dimension $n$, with $r\geqslant nm+n+m$. For every non-negative integer $m$, let $X$ be $m$-osculating regular when its variety of $m$-osculating spaces has dimension $\operatorname{min}\{(m+1)n,r\}$. Suppose that $X$ is $m$-osculating regular. A zero-dimensional scheme is curvilinear if it is locally contained in a smooth curve in $X$.

General infinitesimal Terracini Lemma. $X$ is $m$-defective if and only if, given a general zero-dimensional curvilinear scheme $\gamma$ of length $m+1$ contained in $X$, the dimension of the linear system of hyperplane sections of $X$ singular along $\gamma$ is larger than $r-(m+1)(n+1)$.

This conjecture generalizes the infinitesimal Terracini lemma for secant lines, which is the case $m=1$. It relates defectivity of the secant variety to the failure of the expected number of independent conditions imposed by singularity along a curvilinear scheme; its status is not resolved in the supplied source.