Properness conjecture for Lie triple higher derivations on triangular algebras

Let $\mathfrak{A}=\operatorname{Tri}(\mathcal{A},\mathcal{M},\mathcal{B})$ be a triangular algebra, and let $\mathcal{L}=\{L_n\}_{n\in\mathbb{N}}$ be a Lie triple higher derivation on $\mathfrak{A}$. Properness conjecture. There exists a triangular algebra $\mathfrak{A}^{0}$ containing $\mathfrak{A}$ as a unital subalgebra such that, for every $n\in\mathbb{N}$, $L_n=\Delta_n+\chi_n$, where $\{\Delta_n\}_{n\in\mathbb{N}}$ is a higher derivation with $\Delta_n:\mathfrak{A}\to\mathfrak{A}^{0}$ and $\{\chi_n\}_{n\in\mathbb{N}}$ consists of linear mappings $\chi_n:\mathfrak{A}\to\mathcal{Z}(\mathfrak{A}^{0})$ satisfying $\chi_n([[x,y],z])=0$ for all $x,y,z\in\mathfrak{A}$ and $n\in\mathbb{N}$. This would show that every Lie triple higher derivation is proper after embedding the triangular algebra into a suitable larger triangular algebra. The paper states this as an open problem; no proof or counterexample is given.