Normalizer and conjugator description for standard parabolic subgroups of Artin groups

Let $A$ be an Artin group such that every irreducible $A_Z$ with $|Z|>2$ is not of spherical type. For $X\subseteq S$, write

$$X^\bot=\{y\in S\setminus X\mid xy=yx,\ \forall x\in X\}.$$

If $A_X$ has no cyclic irreducible component $\langle x\rangle$ with $x\in X$, then its normalizer is

$$N(A_X)=A_X\times A_{X^\bot}.$$

Otherwise, the elements that conjugate $X$ to $Y$ are given by

$$\mathrm{Conj}(X,Y)=(\mathrm{Ribb}(X,Y)\cdot A_X)\cdot A_{X^\bot}.$$

This describes normalizers and conjugators of standard parabolic subgroups under the hypothesis that irreducible components with more than two generators are not spherical. The source presents the statement as an auxiliary result adapted from Godelle's conjecture, but provides no resolution status for the underlying claim.