Equivalence of generalized and ordinary Haantjes integrability for commuting operators

Let $M$ be a differentiable manifold and let $\boldsymbol{A},\boldsymbol{B}:TM\rightarrow TM$ be two $(1,1)$ tensor fields. Assume that $\mathcal{H}_{\boldsymbol{A}}(X,Y)=\mathcal{H}_{\boldsymbol{B}}(X,Y)=0$ and $[\boldsymbol{A},\boldsymbol{B}]=0$. Equivalence conjecture.

$$\mathcal{H}_{\boldsymbol{A},\boldsymbol{B}}(X,Y)=0 \Longleftrightarrow \mathcal{H}_{\boldsymbol{A+B}}(X,Y)=0.$$

This conjecture concerns the relationship between the generalized Haantjes binary tensor of two commuting Haantjes operators and the ordinary Haantjes tensor of their sum; the source says that the result is conjectural, without providing a resolution.