  Inspired by recent advances in singular SPDE theory, we use the Poincar\'e
inequality on Wiener space to show that controlled complementary Young
regularity is sufficient to obtain Gaussian rough paths lifts. This allows us
to completely bypass assumptions on the 2D variation regularity of the
covariance and, as a consequence, we obtain cleaner proofs of approximation
statements (with optimal convergence rates) and show the convergence of random
Fourier series in rough paths metrics under minimal assumptions on the
coefficients (which are sharper than those in the existent literature).
