Shifted-binomial log-concavity conjecture for Plücker coefficients

Let $\lambda=(2^{e_2},\ldots,m^{e_m})$ be a partition without parts equal to $1$, and set

$$N:=|\lambda|,\qquad M:=|\widetilde\lambda|,$$

where $\widetilde\lambda$ is obtained by subtracting $1$ from each part. Write the Plücker expansion

$$[\overline Y_\lambda(d)]=\sum_{\lceil M/2\rceil\le i\le M}P_{\lambda;i}(d)s_{i,M-i}.$$

Set $d=N+x$, define $Q_{\lambda;i}(x):=P_{\lambda;i}(N+x)$, and expand

$$Q_{\lambda;i}(x)=\sum_{r\ge0}C_{\lambda;i,r}\binom xr.$$

Shifted-binomial log-concavity conjecture. For every partition $\lambda$ without parts equal to $1$, and every admissible index $i$, the sequence $(C_{\lambda;i,0},C_{\lambda;i,1},C_{\lambda;i,2},\ldots)$ is nonnegative and log-concave. Equivalently, $C_{\lambda;i,r}\ge0$ and

$$C_{\lambda;i,r}^2\ge C_{\lambda;i,r-1}C_{\lambda;i,r+1},$$

with $C_{\lambda;i,r}=0$ outside the degree range of $Q_{\lambda;i}$. The conjecture concerns a shifted binomial refinement of positivity and log-concavity for Plücker coefficients; the source gives no resolution status.