Let $X$ be a Fano 3-fold with terminal singularities. Is there some bound (possibly explicit) for the Picard rank of $X$ ?

If $X$ is smooth, it is well-known that the bound is $10$, obtained by del Pezzo fibrations of degree $1$.

With some assumptions on $X$, the bound is $7$ (Nikulin, "On the Picard number of Fano 3-folds with terminal singularities"). Does someone knows what happens in the general case?

GorensteinFano 3-fold can be deformed to a smooth Fano with the same Picard rank. I don't have access to Namikawa's article, but maybe it's a good place to look for more results in this direction. – Artie Prendergast-Smith Jun 20 '12 at 15:05