= Solution
The measurement cannot definitively exclude the model. If only Standard Model particles supplied entropy at scalar decoupling, the largest available value $g_{*s}=106.75$ would give
$$
\Delta N_{\rm eff}
=\frac47\left(\frac{43}{4\times106.75}\right)^{4/3}
\simeq0.027,
$$
which a $0.1\%$ measurement around the Standard Model value would clearly detect.
In the proposed model, however, the many additional relativistic species are also in equilibrium when the scalar decouples. They increase $g_{*s}(T_d)$, and their later disappearance transfers entropy to the coupled bath but not to the scalar. Since $\Delta N_{\rm eff}\propto g_{*s}(T_d)^{-4/3}$, a sufficiently large hidden particle content can dilute the scalar signal below any stated finite precision; a value above roughly $g_{*s}(T_d)\sim550$ already pushes it below about $0.003$. The null measurement constrains the combination of decoupling time and total entropy degrees of freedom, but does not rule out the entire new-physics model.
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