Alternating-sum decomposition conjecture for the modules $V\!h_k(n)$

Let $Lie^{(2)}$ be the symmetric-function species used in the plethysm below, and let $V\!h_k(n)$ be the $S_n$-module whose Frobenius characteristic is the degree-$n$ term of $h_{n-k}[Lie^{(2)}]$, for $k=0,1,\ldots,n-1$. Define $U_{-1}(n)$ to be the zero module and $U_0(n)$ to be the trivial $S_n$-module. Alternating-sum decomposition conjecture. For $0\leq k\leq n-1$, the truncated alternating sum

$$V\!h_k(n)-V\!h_{k-1}(n)+\ldots+(-1)^kV\!h_0(n)=U_k(n)$$

is a true $S_n$-module, and consequently

$$V\!h_k(n)\simeq U_k(n)+U_{k-1}(n).$$

Equivalently, the degree-$n$ term in

$$(h_{n-k}-h_{n-k+1}+\ldots+(-1)^kh_n)[Lie^{(2)}]$$

is Schur-positive for $0\leq k\leq n-1$. This is an analogue for the modules $V\!h_k(n)$ of the corresponding decomposition of Whitney homology of the partition lattice; the source does not provide a resolution, so the conjecture remains open.