Strong nilpotency criterion for special multi-flag germs

Let a special $m$-flag germ have singularity class

$$\underbrace{{\bf 1 1... 1}}_{l-1}\underbrace{\mathbf{j}_l\mathbf{j}_{l+1}}_{j_l>1}...\mathbf{j}_r.$$

Let $c^i_j$ denote the constants appearing in its EKR normal form. Strong nilpotency conjecture. The germ is strongly nilpotent if and only if

$$c^{l}_{j_l}=\cdots=c_m^l=0,$$ $$c^{l+1}_{j_{l+1}}=\cdots=c_m^{l+1}=0,$$ $$\vdots$$ $$c^{r}_{j_r}=\cdots=c_m^r=0.$$

The vanishing condition is sufficient for strong nilpotency, while the converse remains open, including for Goursat flags.