Alternating expansion conjecture for Young quasisymmetric Schur functions of distinct-part partitions

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ be a partition whose $k$ parts are all distinct. For $\sigma\in S_k$, define

$$\sigma(\lambda)=(\lambda_{\sigma(1)},\lambda_{\sigma(2)},\ldots,\lambda_{\sigma(k)}),$$

and let $\ell(\sigma)$ be the length of $\sigma$, namely the minimum number of adjacent transpositions $(i,i+1)$ needed to generate it. Alternating expansion conjecture. One has

$$\hat{\mathscr{S}}_\lambda=\sum_{\sigma\in S_k}(-1)^{\ell(\sigma)}\mathfrak{S}^*_{\sigma(\lambda)}.$$

The formula gives an explicit signed expansion into dual immaculate quasisymmetric functions for partitions with distinct parts; it is presented as a conjecture based on the authors' computations, and no proof or resolution is supplied here.