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We verify this up to a constant factor for $d$-regular graphs when each edge rings at rate $1/d$ in various cases: (1) when $d = \\Omega( \\log_{n/k} n)$, (2) when $\\mathrm{gap}:=$ the spectral-gap of a single walk is $ O ( 1/\\log^4 n) $ and $k \\ge n^{\\Omega(1)}$, (3) when $k \\asymp n^{a}$ for some constant $0<a<1$. 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