Symmetry conjecture for Littlewood polynomials and equipowerful bipartitions

Let an LP be a Littlewood polynomial, let $n$ be its length, and let $m$ be its exact order. An equipowerful bipartition of length $n$ and exact order $m$ has a witnessing set. A polynomial or witnessing set is $(-1)^m$-symmetric when it has the corresponding symmetry described in the paper.

Symmetry conjecture. If there is an LP of length $n$ and exact order $m$, then there is one that is $(-1)^m$-symmetric. Alternatively, if there is an equipowerful bipartion of length $n$ and exact order $m$, then there is a witnessing set that is $(-1)^m$-symmetric.

All computational evidence in the paper supports this symmetry principle, but the source does not provide a proof or resolution.