All but one expanding Lorenz maps with slope greater than or equal to $\sqrt 2$ are leo

We prove that with only one exception, all expanding Lorenz maps $f:[0,1]\to [0,1]$ with $f’(x)\ge \sqrt 2$ (apart from a finite set of points) are locally eventually onto. Namely, for each such $f$ and each nonempty open interval $J\subset (0,1)$ there is $n\in \mathbb N $ such that $[0,1)\subset f^n(J)$. The exception is the map $f_0(x)=\sqrt 2x+(2-\sqrt 2)/2$ (mod $1$). Recall that $f$ is an expanding Lorenz map if it is strictly increasing on $[0,c)$ and $[c,1]$ for some $c$ and satisfies inf $f’ \gt 1$.