Colombo–Van Geemen nonvanishing conjecture for tautological cycles on Jacobians

Let $C$ be a smooth very general curve of genus $g\geq 2k+1$, and let $JC$ be its Jacobian. Write $Z_k$ for the $k$-th Beauville component of the cycle associated with $C$ in $\operatorname{CH}^{g-1}(JC)_k/\operatorname{alg}$. Colombo–Van Geemen's conjecture.

$$Z_k\neq 0\quad\text{in}\quad \operatorname{CH}^{g-1}(JC)_k/\operatorname{alg}.$$

This asks whether the vanishing theorem for a curve admitting a $g_d^1$ is optimal for a very general curve, whose gonality is $k+2$ when $g=2k+1$ or $g=2k+2$. The conjecture is known for $k=2$ only for general curves of genus at least $11$, whereas the expected bound is genus at least $5$; the general case remains open.