Novak–Rhoades Schur-positivity conjecture for increasing subsequence functions

Let $\Lambda_n$ be the ring of homogeneous symmetric functions of degree $n$ with Kronecker product $*$, and define

$$S_{n,k}=\sum_{\substack{\lambda\vdash n\ell(\lambda)=k}}f^\lambda s_\lambda,$$

where $s_\lambda$ is the Schur function indexed by $\lambda$. Write $F\leq G$ when $G-F$ is Schur positive. Novak–Rhoades' Schur-positivity conjecture. For any $n\geq3$,

$$S_{n,k-1}*S_{n,k+1}\leq S_{n,k}*S_{n,k}$$

for all $2\leq k\leq n-1$. This is another equivalent formulation of the equivariant conjecture under the Frobenius isomorphism; consequently, it would imply the numerical log-concavity conjecture. The source gives no resolution.