# Are weak Fano 4-folds with canonical Gorenstein singularities bounded?

A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big.

Are there finite families of weak Fano 4-folds with canonical Gorenstein singularities? Moreover, in what sense a set of Fano varieties is called "in the same family"?

Any comment on finiteness of Fano varieties are very welcome!

If you further assume $X$ only has canonical singularity, and $-K_X$ is ample, then for any dimension, this is proved in ACC for log canonical thresholds Corollary 1.8.
If you only assume $-K_X$ is big and nef (but still assume $X$ has canonical singularity), from the result above, I believe a standard argument using the finiteness of models (se e.g. [BCHM]) should yield it then.
If you only assume Gorenstein, I doubt it. Can you just take the cone over elliptic curves but with degree higher and higher embedding? I think this give you infinitely many Gorenstein log canonical surfaces with $-K_X$ ample and can't sit in finitely many families.