The twisted and column-even domino-plane-partition bijection conjecture

Let $\GCSPP{n,m}$ be the set of twisted domino plane partitions and let $\CDPP{n,m}$ be the set of restricted domino plane partitions with all columns of even length. Let $\overline U_1$ denote the statistic counting the relevant $1$-parts.

Twisted domino correspondence conjecture. For non-negative integers $m$ and $n\geq1$, there should be a bijection

$$\GCSPP{n,m}\longleftrightarrow\CDPP{n,m}$$

that preserves $\overline U_1$; in particular, the two sets should have the same cardinality and the statistic should have the same distribution.

The conjecture is supported by agreement of computed cardinalities for $n\leq6$. The source does not provide a proof or a general construction of the bijection.