The near-linear value conjecture for cyclic groups

Let $\mathrm{val}(\mathbb Z_n)$ denote the value parameter for the cyclic group $\mathbb Z_n$, as defined in the paper. The near-linear value conjecture. For every $\varepsilon>0$,

$$\mathrm{val}(\mathbb Z_n)\le O(n^{1+\varepsilon}).$$

The paper describes this as its weakest conjecture and uses bounds on $\mathrm{val}(\mathbb Z_n)$ in applications to matrix multiplication. The source does not provide evidence resolving this conjecture, so it remains open.