Rationality and upper bound conjecture for adjacent AKLT edge projectors

Let $S_1$, $S_2$, and $S_3$ be positive integers or half-integers. For adjacent edges $e=\{1,2\}$ and $e'=\{2,3\}$, let $P_{12}=P_{S_1+S_2}(\vec{S}_1+\vec{S}_2)$ and $P_{23}=P_{S_2+S_3}(\vec{S}_2+\vec{S}_3)$, and denote by $s(P_{12}P_{23})$ the largest singular value of $P_{12}P_{23}$ that is not equal to $1$. Rationality and upper bound conjecture. The quantity $s^2(P_{12}P_{23})$ is rational. If $S_1,S_3\leq S_2$, then

$$s^2(P_{12}P_{23})\leq \frac{1}{4},$$

with equality if and only if $S_1=S_2=S_3$. This conjecture concerns the local projector overlaps governing sample-complexity bounds for verification of Affleck–Kennedy–Lieb–Tasaki states. The source notes that the explicit values for spins at most $5/2$ suggest the claim; no resolution is supplied here.