The general existence conjecture for Heffter arrays

Let $H(m,n;s,t)$ denote a Heffter array with $m$ rows, $n$ columns, $s$ filled cells in each row, and $t$ filled cells in each column, so necessarily $ms=nt$. An integer Heffter array is one whose row and column sums are zero over the integers, and a shiftable array is one in which every row and column contains equally many positive and negative entries.

Heffter array existence conjecture. There exists a Heffter array $H(m,n;s,t)$ for all $m,n,s,t$ with $s,t\geq 3$ and $ms=nt$. If the conditions of the necessary integer-sum lemma are satisfied, then there is an integer Heffter array. If the conditions of the necessary shiftability lemma are satisfied, then there is a shiftable array.

This is a broad existence conjecture involving four parameters subject to one equation. Its cases without empty cells, square arrays, and fixed $s$ and $t$ are natural special cases; the general conjecture remains open.