Shifted-binomial log-concavity conjecture for stable CSM–Schur coefficients

Let $\lambda$ be a partition with parts at least $2$, and set

$$c=\operatorname{codim}Y_\lambda=|\lambda|-\ell(\lambda).$$

For every $f\ge c$, write the homogeneous degree-$f$ part of the stable Chern–Schwartz–MacPherson class as

$$c^{SM}(Y_\lambda(d))_f=\sum_{i=0}^{\lfloor f/2\rfloor}C_{\lambda,f,i}(d)s_{f-i,i}(a,b),$$

where $C_{\lambda,f,i}(d)$ is polynomial in $d$ in the stable range. Define

$$\widetilde C_{\lambda,f,i}(d)=(-1)^{f-c}C_{\lambda,f,i}(d),\qquad y=d-|\lambda|-2(f-c).$$

Shifted-binomial log-concavity conjecture. The shifted polynomial

$$\widetilde C_{\lambda,f,i}\bigl(y+|\lambda|+2(f-c)\bigr)$$

has a nonnegative binomial-basis expansion

$$\widetilde C_{\lambda,f,i}\bigl(y+|\lambda|+2(f-c)\bigr)=\sum_{j\ge0}\gamma_{\lambda,f,i,j}\binom yj, \qquad \gamma_{\lambda,f,i,j}\ge0.$$

Moreover, after deleting possible initial zeroes, the coefficient sequence $(\gamma_{\lambda,f,i,0},\gamma_{\lambda,f,i,1},\gamma_{\lambda,f,i,2},\ldots)$ is log-concave:

$$\gamma_{\lambda,f,i,j}^{\,2}\ge\gamma_{\lambda,f,i,j-1}\gamma_{\lambda,f,i,j+1}.$$

The conjecture extends the shifted-binomial positivity and log-concavity pattern from Plücker coefficients to stable CSM classes of coincident-root strata; the source gives no resolution status.