Fredkin-chain conjecture on the central extension by total spin

Let $\mathfrak{g}$ be the Lie algebra generated by the raising and lowering operators of the periodic Fredkin chain with $N$ sites. Let $\mathrm{S}^z$ be the third component of the total spin operator, let $h_i$ be the Cartan subalgebra elements of $\mathfrak{g}$, and set $n=\lceil N/2\rceil$. Central-element conjecture. There exists a nontrivial operator $p$ that is central with respect to $\mathfrak{g}$ and has the form

$$p=\mathrm{S}^z-\sum_{i=1}^{\lceil N/2\rceil}\alpha_i h_i,$$

where the $\alpha_i$ are coefficients. This proposes that the symmetry algebra is extended by a central element obtained from the total-spin component $\mathrm{S}^z$. The claim is based on finite-size computations for $N\leq 10$; determining the coefficients and proving centrality for general $N$ remain open.