Proposition: No commuting duplicate action on the same code

Let $V:L\to \mathcal{H}$ be an isometry. There cannot be two commuting star representations of $\mathcal{B}(L)$ on $\mathcal{H}$ that preserve the code and both satisfy

for every logical $X$, when $d\ge 2$.

Proof. Commutation would give

Injectivity of $V$ would make every pair of logical matrices commute, which is false.