Words with factor somplexity $2n+1$ and minimal critical exponent

We show that words with factor complexity 2n+1 have critical exponent at least $\mu$, where $\mu=2+\frac{1}{\lambda^2-1}= 2.4808726\cdots$, where $\lambda=1.7548777$ is the real zero of $x^3-2x+x-1=0$. This confirms a conjecture of Shallit and Shur.