The secant-syzygy conjecture for Veronese varieties

Let $v_k(\mathbb{P}^d)$ be the $k$-uple Veronese embedding of $\mathbb{P}^d$, with $d\geq2$ and $k\geq3$, and let $\Sigma$ be its secant variety. Write $N_{p,q}$ for the indicated syzygy property. Secant-syzygy conjecture. The secant variety $\Sigma$ satisfies

$$N_{3,3k-5}.$$

This is suggested by computations for $v_3(\mathbb{P}^2)$ and $v_4(\mathbb{P}^2)$, together with known cases for rational normal curves and the preceding conjecture on Veronese syzygies. The source gives no proof or resolution status.