The full Heffter array conjecture

Let $H(m,n)$ denote a Heffter array with no empty cells, so it has $m$ rows and $n$ columns. An integer Heffter array has all row and column sums equal to zero over the integers, while a shiftable array has equally many positive and negative entries in every row and column.

Full Heffter array conjecture. There exist Heffter arrays $H(m,n)$ for all $m,n\geq 3$. Moreover, they are integer Heffter arrays when $mn\equiv 0,3\pmod{4}$, and are shiftable when $m$ and $n$ are both even and at least $4$.

The paper says that the authors believe they have a proof and are preparing details for publication, with computer constructions also available. Thus this special case is presented as conjectural in the source, although the claimed proof was believed by the authors.