Orbit-closure classification of square-free words in odd-difference progressions

Let ${\bf f}=f_0f_1f_2\cdots$ be a paperfolding word over $\{1,4\}$, and define ${\bf v}=v_0v_1v_2\cdots$ by

$$v_{4n}=2,\qquad v_{4n+2}=3,\qquad v_{2n+1}=f_{2n+1}$$

for all $n\geq 0$, with the even-position symbols of ${\bf f}$ recoded as $2$ and $3$. The orbit closure of a word is the set of infinite words obtained as limits of shifts of that word. The orbit-closure classification conjecture. Every infinite word over a 4-letter alphabet that avoids squares in arithmetic progressions of odd difference belongs to the orbit closure of one of the words ${\bf v}$ constructed above. There are uncountably many such words ${\bf v}$ arising from paperfolding words, and the preceding theorem shows that they avoid squares in arithmetic progressions of odd difference; the conjecture asserts that these constructions account for all infinite examples, but no resolution is supplied in the source.