Real-rootedness classification for refined Narayana polynomials

For every pair of integers $1\le k\le n$, consider the refined Narayana polynomial and its formal degree-$k$ symmetric decomposition, written in the form $F_{n,k}(t)=a_{n,k}(t)+t\,b_{n,k}(t)$. The problem is to determine exactly for which pairs $(n,k)$ the polynomials $a_{n,k}(t)$ and the coefficientwise absolute-value polynomial $|b_{n,k}(t)|$ are real-rooted, according to Conjecture 36 of Bóna et al. The supplied source reports that the published classification fails at $(n,k)=(3,3)$: there $a_{3,3}(t)=(1+t)(1+t^2)$ and $|b_{3,3}(t)|=1+t+t^2$, neither of which is real-rooted. A boundary-corrected classification has been proposed, but its general proof has not been independently verified.