Conjecture on termination of the multidimensional Zeckendorf conversion algorithm

Let $k$ be the length of the vector $\vec{\mathbf{c}}=(c_1,\ldots,c_k)$, and let the algorithm in the proof of Theorem main be the conversion algorithm for $\vec{\mathbf{c}}$-representations. A vector is weakly decreasing if $c_{i+1}\leq c_i$ for every $1\leq i<k$. Termination conjecture. If $k=3$, the algorithm terminates when $c_i\geq 0$ for all $1\leq i<3$, $c_{i+1}-c_i\leq 1$ for all $1\leq i\leq k-1$, and $c_k=1$. If there exists $1\leq i<3$ such that $c_{i+1}-c_i\geq 2$, then there is a $\vec{\mathbf{c}}$-representation for which the algorithm fails to terminate. If $k\geq 4$, the algorithm terminates if and only if $\vec{\mathbf{c}}$ is weakly decreasing. The conjecture proposes a precise characterization of when this conversion algorithm terminates; the preceding example shows nontermination for $\vec{\mathbf{c}}=(1,3,1)$, while the asserted general cases remain unresolved in the supplied source.