The fiber-type classification conjecture for path ideals of cycles

Let $C_n$ be the cycle graph on $n$ vertices, and let $I_t(C_n)$ denote its $t$-path ideal. Here, an ideal is of fiber type when its defining ideal is generated by the defining equations of its symmetric algebra together with the defining equations of its special fiber; linear type means that its defining ideal is generated by the linear relations. The notation $t\mid n$ means that $t$ divides $n$.

Fiber-type classification conjecture. (1) If

$$2<t<\left\lfloor\frac{n}{2}\right\rfloor,$$

then $I_t(C_n)$ is of fiber type but not of linear type if and only if $t\mid n$. (2) If

$$\gcd(n,t)>1\quad\text{and}\quad \left\lfloor\frac{n}{2}\right\rfloor<t<n-2,$$

then $I_t(C_n)$ is not of fiber type, and hence not of linear type, except when $n=9$ and $t=6$.

The claim seeks to complete the classification of when path ideals of cycles have fiber type or linear type. The preceding results establish several boundary and parity cases, as well as non-linear-type cases, but the two stated ranges remain the subject of the conjecture.