In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}\mathrm{Sym}^{r}\overline{\mathbb{F}}_{p}^{2}\right)/T$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ is mapped to the irreducible Galois representation $\mathrm{ind}(\omega_{2}^{r+1})$ for $r\in\lbrace 0,\ldots,p-1\rbrace$. See Definition 1.1 in

https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/GL2%28Qp%29II.pdf (with $\eta=1$).

However, I notice that this is not compatible with the local class field theory: on $\mathbb{Z}_{p}^{\times}$, the central character of the supercuspidal representation is the $r$-th power map after going modulo $p$, while the determinant character of the Galois representation is the $(r+1)$-th power map.

So is it defined this way to satisfy some other compatibilities (for example, reduction modulo $p$ of the $p$-adic local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$)?