Quadratic subword-complexity conjecture for the Thue–Morse two-block fixed point

Let $x^{(00)}$ be the fixed point of the Thue–Morse two-block substitution, and let $p^{(00)}(n)$ denote its subword complexity, namely the number of distinct factors of length $n$. Quadratic subword-complexity conjecture. There is a constant $c>0$ such that

$$p^{(00)}(n)\geq cn^2$$

for all $n=1,2,\dots$. The displayed initial values of the complexity function support quadratic growth, but no proof is supplied in the source.