Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense of Vaes-Kustermans). Let us denote its function algebra by $(\ell^\infty(\widehat{\mathbb{G}}), \hat{\Delta})$.

Consider a von Neumann algebra $M$ and a unital completely isometric (normal) map $$\alpha: M \to M \overline{\otimes} \ell^\infty(\widehat{\mathbb{G}})$$ satisfying the coaction property $$(\alpha\otimes \iota) \alpha = (\iota\otimes \hat{\Delta})\alpha.$$ Is it true that $\alpha$ is automatically multiplicative?

Of course, if $\Gamma$ is a discrete group, then a completely isometric map $$\alpha: M \to M\overline{\otimes}\ell^\infty(\Gamma)= \prod_{g\in \Gamma} M$$ is automatically multiplication-preserving because the map $\alpha$ is then a direct product of unital completely isometric maps $\alpha_g: M \to M$ which are automatically $C^*$-isomorphisms (by a result by Choi).

I tried to apply the same trick in this case: $$\ell^\infty(\widehat{\mathbb{G}})\cong \prod_{\gamma \in \operatorname{Irr}(\mathbb{G})} B(H_\gamma)$$ and the map $\alpha$ then breaks down as a collection of maps $$\alpha_\gamma: M \to M_{n_\gamma}(M): m \mapsto [u_{ij}^\gamma \rhd m]$$ where $U^\gamma = [u_{ij}^\gamma]$ is the irreducible representation $\gamma$ and $\rhd: \mathcal{O}(\mathbb{G})\odot M \to M$ the induced left module structure. If we can show that these maps are multiplication-preserving, then we are done. This, in turn, is equivalent with showing that their images are $C^*$-algebras, but neither of these claims are clear to me. On the level of left $\mathcal{O}(\mathbb{G})$-modules, the multiplicativity means $$g\rhd (mn)= (g_{(1)}\rhd m)(g_{(2)}\rhd n)$$ or in terms of matrix coefficients $$u_{ij}^\gamma\rhd (mn) = \sum_{k=1}^{n_\gamma} (u_{ik}^\gamma\rhd m)(u_{kj}^\gamma\rhd n).$$

Is the multiplicativity of the coaction $\alpha$ somehow automatic? I am starting to believe this isn't true, but I was not able to find a counterexample. Thanks in advance for your help!