By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$

Again, following Yarnall's work, we know the spectrum $S^n \wedge H\underline{\mathbb{Z}}$ has the $n$-slice of the form $S^{W(n)}\wedge H\underline{\mathbb{Z}}$. Here $W(n)$ is a certain representation defined in Definition 3.5 of Yarnall's work "The slices of $S^n \wedge H\underline{\mathbb{Z}}$ for cyclic $p$-groups.

$\mathbf{Question:}$ If we take any $C_p$-representation $V=m+n\xi$. Is it true the $\dim(V)$-slice of the spectrum $S^V \wedge H\underline{\mathbb{Z}}$ is of the form $S^{U(m,n)}\wedge H\underline{\mathbb{Z}}$? If so, can we write explicitly what this representation $U(m,n)$ is (may be in terms of $W(m)$ and $n$)?

Thank you so much in advance. Any help will be appreciated.