Spectral-gap and convergence conjecture for SU(2)-symmetric random circuits

Let

$$H=\sum_{i=1}^{n-1}\left(I-T_k^{(i,i+1)}\right)$$

be the Hamiltonian associated with a one-dimensional local random circuit, where $T_k^{(i,i+1)}$ acts on neighboring sites. Let the circuit have open boundary conditions, and let an SU$(2)$-symmetric $2$-design be an ensemble whose second moment approximates the corresponding Haar moment to precision $\epsilon$. Spectral-gap and convergence conjecture. The spectral gap of $H$ scales as

$$\Theta\left(\frac{1}{n^2}\right).$$

Consequently, the 1D local random circuit converges to an $\epsilon$-approximate SU$(2)$-symmetric $2$-design in $\Theta(n^4\log(1/\epsilon))$ steps, whereas the all-to-all interaction random circuit converges in $\Theta(n^3\log(1/\epsilon))$ steps. The conjecture is motivated by numerical results and is presented as requiring rigorous verification.