Good qubit LDPC codes yield good quantum LDPC rotor codes

Let $\phi_\infty(\mathcal{S})$ be a good qubit LDPC code, meaning that it has constant rate and relative distance. Let $p^*$ denote the parameter governing the local-dimension bound, and suppose that

$$p^*<2\pi.$$

Rotor-code conjecture. Under these assumptions, $\phi_\infty(\mathcal{S})$ represents a good quantum LDPC rotor code.

This conjecture proposes that the construction preserves goodness when imported from qubit stabilizer codes to rotor systems, whose two Pauli axes have local dimensions modeled by $\mathbb{R}_{2\pi}^{+}$ and $\mathbb{Z}$. The paper gives the construction under the stated bound but does not provide a proof that the resulting rotor code is good.