Refined Manin conjecture with the canonical exceptional set

Let $X$ be a geometrically uniruled, geometrically integral, smooth projective variety over a number field $F$, and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Assume that $(X,L)$ is adjoint rigid, so that $a(X,L)L+K_X$ has Iitaka dimension $0$. Let $\bold B_+(L)$ be the augmented base locus, let $Z_0=\bold B_+(L)(F)$, and define $Z_1$ and $Z_2$ as the unions of the images $f(Y(F))$ over the respective classes of $F$-thin maps described in the construction: $Z_1$ uses maps with either $\dim Y<\dim X$ or $a(Y,f^*L)f^*L+K_Y$ of positive Iitaka dimension, together with $(a(X,L),b(F,X,L))\leq(a(Y,f^*L),b(F,Y,f^*L))$; $Z_2$ uses dominant face-contracting maps for which $a(Y,f^*L)f^*L+K_Y$ has Iitaka dimension $0$ and the same lexicographic inequality. Refined Manin Conjecture. In Manin's conjecture, assuming $a(X,L)L+K_X$ is rigid, the exceptional set $Z$ can be taken to be

$$Z=Z_0\cup Z_1\cup Z_2.$$

The preceding construction gives a conjectural refinement of the thin exceptional set; the containment of $Z_0\cup Z_1\cup Z_2$ in a thin subset is established in the cited work using the BAB conjecture, but the asserted replacement of $Z$ in Manin's asymptotic remains open.