Palindromicity conjecture for hyperplane sections of high-degree complete intersections

Let $P=\mathbb{P}^N$, and let $X\subset P$ be a general complete intersection of multi-degree $(d_1,d_2,\ldots,d_k)$ with $d_1\leq\cdots\leq d_k$. Assume that $d_1\gg0$. For a hyperplane $H\in P^{\vee}$, write $Y:=X\cap H$. Palindromicity conjecture. The hyperplane section $Y$ is palindromic. This conjecture generalizes the preceding result for hyperplane sections of general cubic $4$-folds, and asks whether palindromicity persists for hyperplane sections of general complete intersections when the smallest defining degree is sufficiently large.