The signed left Pieri rule for the immaculate basis

Let $m$ be a nonnegative integer and let $\alpha$ be a composition. Write ${H}_m$ for the noncommutative complete homogeneous function, ${\mathfrak S}_\alpha$ for the immaculate basis element indexed by $\alpha$, and let $|\alpha|$ denote the size of $\alpha$. A collection of compositions $\beta$ of size $|\alpha|+m$ is called the indexing collection, and $sign(\alpha,\beta)$ is a statistic on pairs of compositions.

The signed left Pieri rule. The product has the form

$${H}_m {\mathfrak S}_\alpha=\sum_\beta(-1)^{sign(\alpha,\beta)}{\mathfrak S}_\beta,$$

where the sum is over some collection of compositions $\beta$ of size $|\alpha|+m$.

This conjecture proposes that left multiplication by a complete homogeneous function is multiplicity free up to signs, in contrast with the positive right Pieri rule. The source does not specify the indexing collection or statistic, and gives no resolution status.