Zero-multiplicity conjecture for the k-generalized Fibonacci sequence with non-positive indices

Let $H^{(k)}$ be the $k$-generalized Fibonacci sequence with non-positive indices, and let $\mathcal{Z}(H^{(k)})=\{n\in\mathbb{Z}:H_n^{(k)}=0\}$ denote its zero set. The zero-multiplicity conjecture. For every $k\geq4$,

$$\#\mathcal{Z}(H^{(k)})=\frac{k(k-1)}{2}.$$

This asserts that the zero-multiplicity is the $(k-1)$st triangular number. The claimed value agrees with the previously established lower bound and with computations for $4\leq k\leq500$, but the conjecture remains open in the supplied source.