Shifted Schur Q-function lattice positivity conjecture

Let $\lambda/\mu$ and $\nu/\rho$ be skew shifted shapes. For partitions, define componentwise operations

$$\lambda\vee\nu=({\rm \max}(\lambda_1,\nu_1),{\rm \max}(\lambda_2,\nu_2),\ldots),\qquad \lambda\wedge\nu=(\min(\lambda_1,\nu_1),\min(\lambda_2,\nu_2),\ldots),$$

and extend them to skew shifted shapes by $(\lambda/\mu)\vee(\nu/\rho)=(\lambda\vee\nu)/(\mu\vee\rho)$ and $(\lambda/\mu)\wedge(\nu/\rho)=(\lambda\wedge\nu)/(\mu\wedge\rho)$. Shifted Schur $Q$-function lattice positivity conjecture. The difference

$$Q_{(\lambda/\mu)\vee(\nu/\rho)}Q_{(\lambda/\mu)\wedge(\nu/\rho)}-Q_{\lambda/\mu}Q_{\nu/\rho}$$

is a non-negative combination of Schur $Q$-functions. The source notes that the analogous difference is already known to be a nonnegative sum of peak functions, while this stronger Schur $Q$-positivity assertion remains conjectural.