I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple of such mathematical phenomena that might be more familiar.

Poincaré's conjecture (now a theorem) in dimension $3$ persisted much longer than in higher dimensions.

Congruence modulo primes for the partition function $p(n)$ lingers for primes $p=2, 3$ while a recent work on Maass forms settles such for higher primes.

Hoping that these citations shed light, I like to ask:

QUESTION.Do you know of conjectures (problems) which manifested to be either notoriously harder or unsolved for "lower dimensions/orders/primes" compared to their "higher dimensional/order/prime" cousins?

Poincaré conjecture persisted in higher dimension": this depends on the way it's formulated. "every simply connected compact smooth $d$-manifold is homeomorphic to the $d$-sphere": false in each dimension $\ge 4$; "every contractible compact smooth $d$-manifold is homeomorphic to the $d$-sphere": true in every dimension $\ge 4$; "every contractible compact smooth $d$-manifold is diffeomorphic to the $d$-sphere": open in dimension 4, true in dimension 5, 6, 12, false in most dimensions $\ge 7$ including all large enough dimensions. $\endgroup$ – YCor May 9 at 22:05having the homotopy type of the $d$-sphere". (Maybe for a closed connected $d$-manifold it's equivalent assuming vanishing of $\pi_i$ for $i<d$.) $\endgroup$ – YCor May 10 at 6:00