Cone of a Fano variety and klt type singularities

Question

Let $X$ be a smooth Fano variety, $\mathcal{L}$ an ample line bundle on $X$, and let $C:=C(X,\mathcal{L})$ be the ample cone. If $\mathcal{L}=\mathcal{O}_X(-mK_X)$, then $K_C$ is $\mathbb{Q}$-Cartier, and $C$ has klt singularities (cf. [Kol13, Lemma 3.1]). However, if $\mathcal{L}$ is not proportional to $K_X$, then $K_C$ is never $\mathbb{Q}$-Cartier.

There is a notion of of klt type. Indeed, $X$ is of klt type if there is an effective $\mathbb{Q}$-Cartier divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt, and it can be a suitable notion of klt in the setting of non $\mathbb{Q}$-Gorenstein case. The question is,

For every smooth Fano variety $X$ (or more generally, a klt log Fano pair $(X,\Delta)$) over $\mathbb{C}$, and an ample line bundle $\mathcal{L}$ on $X$, is the ample cone $C:=C(X,\mathcal{L})$ of klt type?

Let us consider a positive characteristic analog of the question:

For every projective globally $F$-regular variety $X$ over a perfect field, and an ample line bundle $\mathcal{L}$ on $X$, if the ample cone $C:=C(X,\mathcal{L})$ globally $F$-regular?

The answer is yes by [SS10, Proposition 5.3]. In [SS10, Remark 5.5], they considered a similar problem, but they did not answer or question my problem. Can we prove the problem using [BCHM10] or other technique? Note that if [SS10, Question 7.1] is true, then since $C$ is of globally $F$-regular type, $C$ is of klt type, and hence we may regard our problem as a corollary of a conjecture posed by Schwede and Smith.

[BCHM10] C. Birkar, P. Cascini, C. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405-468.

[Kol13] J. Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, (2013).

[SS10] K. Schwede and K. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863--894.

Answer 1

The answer to your first question is YES. You actually do not need any technique other than the definition.

If you check [Kol13, Lemma 3.1], you will find that if $-(K_X+B)\sim_\mathbb{Q} rL$ for some $r>0$ and klt pair $(X, B)$, then $C(X, L)$ is of klt type. Then it suffices to construct such a pair $(X,B)$.

Suppose that $(X, \Delta)$ is log Fano, then $-(K_X+\Delta)-r L$ is ample for sufficiently small $r>0$. Then we can find $0\leq B'\sim_\mathbb{Q}-(K_X+\Delta)-r L$ such that $(X, \Delta+B')$ is klt. Then take $B=\Delta+B'$.