Exactly two standard words in the finite corank-two residue

The independent exactly-two assertion of Dyer–Gimenez §1.27 applies to $r=\nu (T\setminus I,I)$, since $\mathrm{\mid }T\setminus I\mathrm{\mid }=2$ and the finite-relative-root hypothesis is exactly that of Lemma 1.25. It gives precisely two minimum generator words for $r$. By Standard expressions are minimum generator words, inversion and reversal give precisely two minimum generator words for $q$, and every standard expression of $q$ belongs to this set. This uses only the word-count assertion of §1.27, not its incorrectly indexed rightmost-factor clause.

By Elementary inversions detect the two initial prefixes, $q=v_{c,I}{z}_c$ with lengths additive, for each $c=a,b$. Write $v_{c,I}:I\to I_c$ and $q:I\to K$. The root-action condition in Standard expressions and their lengths gives $q{\mathrm{\Pi }}_K={\mathrm{\Pi }}_I$ and $v_{c,I}{\mathrm{\Pi }}_{I_c}={\mathrm{\Pi }}_I$. Hence $z_c{\mathrm{\Pi }}_K=v_{c,I}^{-1}q{\mathrm{\Pi }}_K=v_{c,I}^{-1}{\mathrm{\Pi }}_I={\mathrm{\Pi }}_{I_c}$, so $z_c:I_c\to K$ is an article morphism, composable after $v_{c,I}$. By Every standard expression counts the same parallel classes, $z_c$ has a standard expression. Prepending $v_{c,I}$ gives a composable expression for $q$ whose Coxeter lengths add, so it is standard by Standard expressions and their lengths. The two words are distinct because $v_{a,I}\mathrm{\ne }{v}_{b,I}$ by the elementary-prefix lemma. There can be no others by step 1.1. Their factor counts agree by Brink–Howlett generators and invariance of standard length.