Special-values conjecture for coefficients of cap-shaped link patterns

Let $\mathcal{D}_n$ be the set of link patterns of size $n$, and let $c_{\alpha\pi}$ be the coefficients introduced in the paper. Write link patterns as binary words, with $\mathbf{1}_i=(01)^i$ and $\mathbf{0}_{\ell+1}=0^{\ell+1}1^{\ell+1}$. Suppose

$$\pi=(01)^i0^{\ell+1}1^{\ell+1}(01)^j=\mathbf{1}_i\mathbf{0}_{\ell+1}\mathbf{1}_j.$$

Special-values conjecture. The coefficient is

$$c_{\alpha\pi}=\begin{cases} 1&\text{if $\alpha=u\mathbf{0}_{\ell}v$ with $|u|=2i+1$ and $|v|=2j+1$,}\\ 0&\text{otherwise.} \end{cases}$$

When $\ell=0$, this specializes to Thapper's conjecture. The paper states that the conjecture would imply a relation between numbers of fully packed loop configurations on the square grid; its status is not resolved in the supplied text.