= Solution
The concern is that the rapid-decay assumptions need not be preserved by the dynamics. In a constant-density <incompressible flow>, the divergence of the <Navier-Stokes equation> gives the <pressure Poisson equation for incompressible flow>, with $\pi=p/\rho$:
$$
\nabla^2\pi=-\partial_i\partial_j(u_iu_j).
$$
For a localized eddy, inversion with the free-space <Green function> and two integrations by parts yield
$$
\pi(\boldsymbol r)=\frac1{4\pi}\int u_i(\boldsymbol y)u_j(\boldsymbol y)
\partial_{y_i}\partial_{y_j}\frac1{|\boldsymbol r-\boldsymbol y|}d^3y.
$$
At large separation the differentiated kernel has leading form $(3n_in_j-\delta_{ij})/r^3$. Thus an anisotropic stress moment of an individual eddy produces a quadrupolar <fluid pressure> of order $r^{-3}$, even if the <velocity> source is spatially localized. Correlating that <fluid pressure> with local $u_x^2$ allows $\langle u_x^2p'\rangle=O(r^{-3})$. Isotropy may cancel a mean multipole but does not automatically cancel such a mixed fourth-order statistic.
The triple-correlation evolution contains a pressure-gradient term of the form $-\langle u_x^2\partial_{x'}\pi'\rangle$, so these <long-range pressure correlations in turbulence> can generate a longitudinal triple-correlation tail $K\sim ar^{-4}$. With $L=0$, part (ii) then gives $\dot I=8\pi u^3a$, which need not vanish. This is the substance of the objection to deducing exact conservation from initially short <velocity> correlations. Batchelor and Proudman's original long-range calculation concerned anisotropic turbulence; extending it to isotropic decay requires the appropriate higher-order correlations and allows cancellations. A permitted tail is not a proof that its coefficient stays appreciable in fully developed turbulence.
This possibility appealed to workers constructing heuristic <turbulence closures>. A closure that generated a changing low-wavenumber $k^4$ coefficient and a decay exponent different from $10/7$ could then be viewed as representing a pressure-mediated effect, rather than as automatically violating a universal invariant. The closure's far-field coefficient still needs physical validation; closure-generated nonconservation does not itself prove the true dynamics nonconserving.
For evidence available before this exam, https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/on-the-decay-of-isotropic-turbulence/72E017B53646320E97C378A2F6B94C0E[the 2006 simulations of Ishida, Davidson and Kaneda] found that $I$ approached an approximately constant value after a transient when the box was much larger than the <integral scale of turbulence> and the <Reynolds number> was large. The energy decay was correspondingly close to $u^2\sim t^{-10/7}$. \b[The observed conclusion was approximate late-time conservation in that regime], indicating weak net long-range triple-correlation transfer, rather than exact conservation for every initial condition, box size or Reynolds number.
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