Is there a CSS code for which the controlled phase gate and all Clifford gates are transversal?

In a paper that I have recently read, a protocol was given that required a CSS code with certain properties for which all logical Clifford gates and standard measurements have a transversal implementation (the 7-qubit would be an example). Later in that same paper a procedure was required that required the controlled $XP^{dagger}$ ($C-XP^{dagger}$) gate to be transversal for that specific code.

Now my question is, is that even possible? As far as I know, the set {Clifford group + a single non-Clifford} gate is universal, and that no stabilizer code allows for transversal implementation of a universal set of gates without any further assumptions. The controlled phase gate $C-P$ is a non-Clifford gate and $C-XP^{dagger} = CNOT*(C-P^{dagger})$, so that $C-XP^{dagger}$ also is a non-Clifford gate. Doesn’t this make the requirement that the ($C-XP^{dagger}$) gate be transversal impossible in the first place?