Positive-characteristic groupwise isomorphism conjectures for universal enveloping algebras

Let $L$ and $H$ be Lie algebras over a field, and let $U(L)$ and $U(H)$ denote their universal enveloping algebras. The paper partitions the relevant four-dimensional solvable Lie algebras into Groups 2, 5, and 6.

Groupwise positive-characteristic isomorphism conjectures. The following three statements should hold:

1. If the field has positive characteristic, $L$ belongs to Group 5, $H$ belongs to Group 6, and $U(L)\cong U(H)$, then $L\cong H$.
2. If the field has characteristic two, $L$ and $H$ belong to Group 2, and $U(L)\cong U(H)$, then $L\cong H$.
3. If the field has positive characteristic, $L$ and $H$ belong to Group 5, and $U(L)\cong U(H)$, then $L\cong H$.

These are the three statements identified by the paper as sufficient to extend its characteristic-zero isomorphism theorem to arbitrary characteristic. The paper does not verify them.