Conjecture on PG-shell Lefschetz chains for projective embeddings

Let $X$ be a complex projective manifold of dimension $n>0$, let $j:X\hookrightarrow P=\mathbb{P}^{N}(\mathbb{C})$ be an arithmetically normal embedding, and let $\{W_p\}_{p=0}^n$ and $\{W_p^*\}_{p=0}^n$ be suitably chosen Lefschetz and dual Lefschetz chains of $j(X)$. A PG-shell Lefschetz-chain conjecture asserts that the chains can be chosen so that each $W_p$ and $W_p^*$ is a PG-shell of $j(X)$, reduced along $j(X)$, and irreducible; each restricted conormal sheaf

$$N^{\vee}_{W_p/W_{p-1}}|_X$$

is a vector bundle on $X$ extendable to $W_{p-1}$ as a vector bundle, and likewise

$$N^{\vee}_{W_p^*/W_{p+1}^*}|_X$$

is a vector bundle extendable to $W_{p+1}^*$ as a vector bundle. Moreover, for a fixed manifold $X$ of dimension $n\geq 2$, one can suitably choose the embedding and the two chains so that a refinement of either chain realizes the Working Hypothesis referred to in the source.