Alternating-sign and leading-term conjecture for exceptional-degree pushforwards

Let $X=G/P_X$ be a cominuscule flag variety, let $u,v\in W^X$, and assume that $d=d_{\max}(u^\vee,v)+1$ is an exceptional degree of ${\mathcal O}_u\star{\mathcal O}^v$. Let $p_d:Z_{d-1,1}(X_u,X^v)\to\Gamma_d(X_u,X^v)$ be the morphism from the space of degree-$d$ stable-map data, and let $\operatorname{lead}({\mathcal F})$ denote the lowest-degree component of the Chern character of a nonzero class ${\mathcal F}\in K(X)$. Let $\operatorname{codim}({\mathcal F})$ be the complex degree of this initial term. A class has alternating signs when $(-1)^{\ell(w)-\operatorname{codim}({\mathcal F})}c_w({\mathcal F})\geq0$ for every Schubert-basis coefficient $c_w({\mathcal F})$.

Exceptional-degree pushforward conjecture. Under these assumptions:

1. The class

$$(p_d)_*[{\mathcal O}_{Z_{d-1,1}}(X_u,X^v)]\in K(X)$$

has alternating signs. 2. If

$$\dim\Gamma_d(X_u,X^v)\not\equiv\dim Z_d(X_u,X^v)\pmod 2,$$

then

$$\operatorname{lead}((p_d)_*[{\mathcal O}_{Z_{d-1,1}}(X_u,X^v)])=2[\Gamma_d(X_u,X^v)].$$

3. If

$$\dim\Gamma_d(X_u,X^v)\equiv\dim Z_d(X_u,X^v)\pmod 2,$$

then the initial term of this pushforward has complex degree

$$\operatorname{codim}(\Gamma_d(X_u,X^v),X)+1.$$

Parts (b) and (c) imply the nonvanishing of the degree-$d$ quantum $K$-theory component at every exceptional degree. The first part is motivated by a possible generalization of Brion's positivity theorem; the supplied text gives no resolution.