conjectured that if $A$ is an ample line bundle on a complex abelian variety $X$, then the line bundle $A^{\ell}$ satisfies property $N_p$ whenever $\ell {\geqslant}p + 3$, [@Laz:vb Conjecture 1.5.1, p. 516]. This conjecture was proven by Pareschi, see [@Par], and then later improved by Pareschi-Popa. More specifically, using the fact that the line bundle $A^{\ell}$ satisfies property $N_p$ when $\ell {\geqslant}p + 3$, Pareschi-Popa proved that the line bundle $A^{\ell}$ satisfies property $N_p$ when $\ell = p + 2$, [@Par:Popa:II Theorem 6.2, p. 184].

Central to Pareschi’s proof, and Pareschi-Popa’s refinement, of Lazarsfeld’s conjecture is the study of cup-product maps of the form $${\mathrm{H}}^0(X, A^{n}) \otimes {\mathrm{H}}^0(X, E) \xrightarrow{\cup} {\mathrm{H}}^0(X, A^{n} \otimes E)$$ where $A$ is a globally generated ample line bundle on $X$ and where $E$ is a suitably defined vector bundle on $X$.

To study these maps, Pareschi-Popa formulated *index conditions*, for instance the IT, WIT, and PIT [@Par p. 653] and [@Par:Popa:II p. 179], which were motivated by those of S. Mukai [@Mukai-duality p. 156]. They used these conditions to study families of cup-product maps of the form $$\label{P-P-family-cp}
{\mathrm{H}}^0(X, T^*_x(A^{ n})) \otimes {\mathrm{H}}^0(X, E) \xrightarrow{\cup} {\mathrm{H}}^0(X, T^*_x(A^{ n}) \otimes E)$$ as $x \in X$ 