Structural characterization of 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 sequence of $\mathcal{R}$-linear mappings on $\mathfrak{A}$. Structural characterization conjecture. The sequence $\mathcal{L}$ is a Lie triple higher derivation on $\mathfrak{A}$ if and only if, for each $n\in\mathbb{N}$, $L_n$ has the displayed matrix representation with linear mappings $f_n$, $p_n$, $h_n$, $q_n$, and $g_n$ satisfying conditions (i)--(iii) in the source statement. Here $\mathbb{N}$ is the set of nonnegative integers, $\mathcal{M}$ is the bimodule in the triangular algebra, and $[\mathcal{A},\mathcal{A}]'$ and $[\mathcal{B},\mathcal{B}]'$ are the respective commutator-related subspaces appearing in the source. This would characterize Lie triple higher derivations through their diagonal and off-diagonal components, extending structural results for related derivations on triangular algebras. The problem is posed as an open problem in the paper, and no resolution is supplied.