Weak combinatorial invariance conjecture for dominant affine Bruhat intervals

Let $W=\mathbbm{Z}\Phi^\vee\rtimes W_f$ be an affine Weyl group, let $W_+$ be the set of dominant elements, and let $\Lambda^\vee$ be the coweight lattice. Let $G$ be the group of Bruhat-order automorphisms generated by inversion and Dynkin-diagram symmetries. For $x,y\in W$, write $y-x\in\Lambda^\vee$ when $y=x+\lambda$ for some $\lambda\in\Lambda^\vee$. An interval is full as in the source's affine Bruhat-interval terminology.

Weak combinatorial invariance conjecture. If $x,x',y,y'\in W_+$ satisfy that $[x,y]$ and $[x',y']$ are full and poset-isomorphic, and there exists $g_0\in G$ with $x-g_0x'\in\Lambda^\vee$, then there exists $g\in G$ such that

$$y-gy'=x-gx'\in\Lambda^\vee.$$

The conjecture is motivated by computational evidence and is stated for arbitrary affine Weyl groups. The source notes that it is established for thick intervals in type $\widetilde{A}_2$, but gives no general resolution.