Effective linearization conjecture for configurations of linear subspaces

Let $V$ be an $n$-dimensional complex vector space, let $X=\prod_{i=1}^m \operatorname{Gr}(k_i,V)$, and let $G=\operatorname{SL}(V)$ act on $X$. For a weight vector ${\bf \omega}=(\omega_1,\ldots,\omega_m)$, let $L_{\bf \omega}$ be the corresponding linearized ample line bundle, and write $X^{ss}(L_{\bf \omega})$ and $X^s(L_{\bf \omega})$ for its semistable and stable loci. Assume that $G$ acts freely on a generic configuration of linear subspaces, as above, possibly subject to additional natural conditions.

Effective linearization conjecture.

$$X^{ss}(L_{\bf \omega}) \ne \emptyset$$

if and only if

$$\omega_i\leq {1\over n}\sum_i k_i\omega_i$$

for every $1\leq i\leq m$, equivalently

$$\max_i\omega_i\leq {1\over n}\sum_i k_i\omega_i.$$

Moreover,

$$X^s(L_{\bf \omega}) \ne \emptyset$$

if and only if

$$\omega_i<{1\over n}\sum_i k_i\omega_i$$

for every $1\leq i\leq m$, equivalently

$$\max_i\omega_i<{1\over n}\sum_i k_i\omega_i.$$

The necessary parts of both assertions are proved in the paper; the sufficiency under the stated hypotheses remains conjectural. The criterion is intended to characterize the effective ample linearizations and the corresponding nonempty semistable and stable loci.