Conjecture on the Stanley depth of cyclic graph edge ideals

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring, and let $J_n$ denote the ideal associated with the cyclic graph on $n$ vertices as in the paper. The Stanley depth conjecture. For all $n\geq 10$ satisfying $n\equiv 1\pmod{3}$,

$$\operatorname{sdepth}(S/J_n)=\left\lceil\frac{n}{3}\right\rceil.$$

This conjecture concerns the remaining congruence class in the authors' study of the Stanley depth of quotients by ideals associated with line and cyclic graphs; the stated examples support the formula, but the claim is not resolved in the supplied source.