Wang–Zhang–Zhang's immanant conjectures for cyclic groups

Let $C_n$ be the cyclic group of order $n$. For a partition $\lambda$ of $n$, let $\mathcal I_\lambda(C_n)$ denote the number of formally different monomials occurring in the immanant of the Cayley-table matrix of $C_n$. Write $\mathcal P(C_n)$ and $\mathcal D(C_n)$ for the corresponding quantities for the permanent and determinant, respectively.

Wang–Zhang–Zhang's immanant conjectures.

1. For any $n ge 3$, if $n$ is odd, then

$$\mathcal I_{(n-1,1)}(C_n)=\mathcal I_{(2,1^{n-2})}(C_n)=0.$$

2. For any $n ge 2$, if $nequiv 2\pmod 4$, then

$$\mathcal I_{(n-1,1)}(C_n)=\mathcal P(C_n),\qquad \mathcal I_{(2,1^{n-2})}(C_n)=\mathcal D(C_n).$$

3. For any $n ge 7$, if $n$ is odd, then

$$\mathcal I_{(n-2,1,1)}(C_n)=\mathcal P(C_n).$$

These conjectures extend questions about when immanant monomial counts agree with the permanent or determinant counts, and concern the behavior of these quantities for cyclic groups in the odd and $2$ modulo $4$ cases. The supplied text does not establish their status beyond presenting them as conjectures.