The Whitehouse-type lifting conjecture for $Lie_n^{(2)}$

Let $Lie_n^{(2)}$ denote the relevant $S_n$-module, and let $Lie_{n-1}^{(2)}\uparrow^{S_n}$ denote induction from $S_{n-1}$ to $S_n$. The Whitehouse-type lifting conjecture. The symmetric function $p_1Lie_{n-1}^{(2)}-Lie_n^{(2)}$ is Schur-positive if and only if $n$ is not a power of $2$. Equivalently,

$$Lie_{n-1}^{(2)}\uparrow^{S_n}-Lie_n^{(2)}$$

is a true $S_n$-module lifting $Lie_{n-1}^{(2)}$ if and only if $n$ is not a power of $2$. The source reports verification by computation through $n=32$, but gives no proof or disproof.