Schur positivity of difference of products of derived Schur polynomials

To any Schur polynomial $s_{\lambda }$ one can associated its derived polynomials $s_{\lambda }(i)$ $i=0,\dots ,\mathrm{\mid }\lambda \mathrm{\mid }$ by the rule $$s_{\lambda }(x_1+t,\dots ,x_n+t)=\sum _i{s}_{\lambda }^{(i)}(x_1,\dots ,x_n)t^i\mathrm{.}$$ We conjecture that $$(s_{\lambda }^{(i)}{)}^2-s_{\lambda }^{(i-1)}{s}_{\lambda }^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $\lambda =(k^{\mathrm{\ell }})$, for hooks $\lambda =(k,1^{\mathrm{\ell }-1})$, and when $\lambda =(k,k,1)$ or $\lambda =(3,2^{k-1})$.