Decorrelation of $\mathrm{S}\mathrm{O}(5)\times \mathrm{S}\mathrm{O}(2)$ Bessel periods for symmetric cubes of $\mathrm{G}\mathrm{L}(2)$

We demonstrate that for symmetric cubes of algebraic regular Hecke eigenforms on $\mathrm{G}\mathrm{L}(2)$, a decorrelation phenomenon occurs for global Bessel periods of $\mathrm{S}\mathrm{O}(5)\times \mathrm{S}\mathrm{O}(2)$ averaged over imaginary quadratic fields, conditional on the Generalized Riemann Hypothesis.