Uniform periodic counterexamples to Carleson's convergence problem with polynomial symbols

In Carleson's convergence problem for dispersive equations $i\, \partial_t u + P(D)u=0$ in the periodic setting $\mathbb T^d$, we prove that the Sobolev exponent $d/(2(d+1))$ is necessary for any non-singular polynomial symbol $P$, including the natural powers of the Laplacian $\Delta^k$. This is in contrast with the results known in the Euclidean case, in which for symbols $P(\xi) = |\xi|^a$ with $a > 1$ the exponent $d/(2(d+1))$ is sufficient, but we do not know if it is necessary.