The statistical model of compound-nucleus reactions predicts that the fluctuations of the partial $gamma$-decay widths for a compound-nucleus resonance are governed by the Porter-Thomas distribution (PTD), and that consequently the distribution of total $gamma$-decay widths is very narrow. However, a recent experiment [Koehler, Larsen, Guttormsen, Siem, and Guber, Phys. Rev. C 88, 041305(R) (2013)] reported large fluctuations of the total $gamma$-decay widths in the $^{95}$Mo$(n,gamma)^{96}$Mo* reaction, contrary to this expectation. Furthermore, in recent theoretical works it was argued that sufficiently strong channel couplings can cause deviations of the partial width distributions from PTD. Here, we investigate whether the combined influence of a large number of nonequivalent $gamma$-decay channels, each of which couples weakly to the compound-nucleus resonances, can modify the statistics of the partial widths. We study this effect in neutron scattering off $^{95}$Mo within a random-matrix model that includes coupling to the entrance neutron channel and to the large number of $gamma$ channels. Using realistic coupling parameters obtained from empirical models for the level density and the $gamma$ strength function, we find that the PTD describes well the distribution of partial widths for all decay channels, in agreement with the statistical-model expectation. Furthermore, we find that the width of the distribution of the total $gamma$-decay widths is insensitive to wide variations in the parameters of the $gamma$ strength function, as well as to deviations of the partial-width distributions from the PTD. Our results rule out an explanation of the recent experimental data within a statistical-model description of the compound nucleus.
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