The iSWAP equality conjecture for complete-graph gadgets

Let $K_n$ be the complete graph on $n$ vertices, and consider a gadget associated with a family of 2-local gates having parameter $a=\frac{5}{9}$ in the notation of the paper. Let $\operatorname{\mathcal{T}}_{2,K_n}^{\operatorname{IG}}$ denote the corresponding second-moment operator for the gadget family, and let $\Delta(\cdot)$ denote the convergence gap. The iSWAP equality conjecture. For $n\geq 5$,

$$\Delta\!\left(\operatorname{\mathcal{T}}_{2,K_n}^{\operatorname{IG}}\right)=\Delta\!\left(\operatorname{\mathcal{T}}_{2,K_n}^{\operatorname{\mathsf{iSWAP}}}\right).$$

In particular, gadgets associated with $\operatorname{\mathsf{iSWAP}}$, $\operatorname{\mathsf{B}}$, or $\operatorname{\mathsf{CNOT}}$ satisfy the stated parameter requirement and are conjectured to attain the fastest convergence toward unitary 2-designs for sufficiently large $n$. The paper explains that a stronger spectral statement would imply the conjecture, but that a refined method is needed for a proof.