Compatibility of the rigged-configuration bijection with non-affine Kashiwara operators

Let $\Phi$ be the map from rigged configurations of a tensor product of type $D_n^{(1)}$ Kirillov--Reshetikhin crystals to the corresponding crystal, and let $e_i$ and $f_i$ denote the Kashiwara operators for $i\neq 0$. Operator-compatibility conjecture. The map $\Phi$ commutes with these operators:

$$\Phi\circ e_i=e_i\circ\Phi,\qquad \Phi\circ f_i=f_i\circ\Phi \quad (i\neq 0).$$

The analogous statement for type $A$ was proved by Schilling and collaborators, and the authors expect similar methods to apply in type $D$; no resolution is supplied here.