Non-power-of-two lifting conjecture for $Lie_n^{(2)}$

Let $Lie_n^{(2)}$ denote the $S_n$-representation considered in the paper, and let $p_1Lie_{n-1}^{(2)}-Lie_n^{(2)}$ be the corresponding symmetric function; equivalently, let $Lie_{n-1}^{(2)}\uparrow^{S_n}-Lie_n^{(2)}$ denote the induced virtual representation. Non-power-of-two 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 $2$. Such a lifting would generalize the Whitehouse-module construction for the ordinary Lie representation; the conjecture was verified computationally through $n=32$, but no general proof is given.