Do high-order structure-function exponents saturate at large order?

In isotropic Navier-Stokes turbulence at $\mathrm{Re}_{\lambda} \to \infty$, do the transverse and longitudinal exponents $\zeta_p^T$ and $\zeta_p^L$ tend to a common finite limit $\zeta_{\mathrm{inf}}$ as $p \to \infty$? DNS up to $\mathrm{Re}_{\lambda} \sim 1300$ report saturation of transverse exponents near $\zeta_{\mathrm{inf}} \sim 2$. Answer: yes or no for each increment type, with $\zeta_{\mathrm{inf}}$ and error bars from statistically converged tails for $p$ up to about 12.

Converged DNS or experimental moments of both increment types at $Re_{\lambda}$ above about 2000 showing a common plateau, or a demonstrated difference.