The fine-set conjecture for vertically rotated one-column grid classes

Let $\mathbf{v}=(v_1,\dots,v_r)\in\{1,-1\}^r$, and let ${\mathcal G}^{\mathbf{v}}$ be the one-column grid class with slope vector $\mathbf{v}$. Write $C_n$ for the $n$-cycle and ${\mathcal G}_n^{\mathbf{v}}$ for the permutations of size $n$ in this grid class.

Fine-set conjecture. For every one-column grid class ${\mathcal G}^{\mathbf{v}}$, the set

$$\{C_n{\mathcal G}_n^{\mathbf{v}}\}$$

is a fine set.

The claim generalizes the cases where all slopes have the same sign and the cases $\mathbf{v}=-+$ and $\mathbf{v}=+-$, which are established in the paper. It is motivated by computer experiments; the general case remains open.